Properties

Label 4-84e2-1.1-c7e2-0-2
Degree $4$
Conductor $7056$
Sign $1$
Analytic cond. $688.555$
Root an. cond. $5.12253$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 54·3-s + 264·5-s + 686·7-s + 2.18e3·9-s − 4.98e3·11-s − 1.01e4·13-s − 1.42e4·15-s + 1.78e4·17-s + 6.25e3·19-s − 3.70e4·21-s + 1.40e4·23-s + 2.73e4·25-s − 7.87e4·27-s + 2.43e5·29-s + 4.70e5·31-s + 2.68e5·33-s + 1.81e5·35-s + 3.11e5·37-s + 5.47e5·39-s + 9.19e5·41-s − 1.12e5·43-s + 5.77e5·45-s − 1.02e5·47-s + 3.52e5·49-s − 9.62e5·51-s + 2.72e6·53-s − 1.31e6·55-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.944·5-s + 0.755·7-s + 9-s − 1.12·11-s − 1.28·13-s − 1.09·15-s + 0.880·17-s + 0.209·19-s − 0.872·21-s + 0.240·23-s + 0.350·25-s − 0.769·27-s + 1.85·29-s + 2.83·31-s + 1.30·33-s + 0.713·35-s + 1.00·37-s + 1.47·39-s + 2.08·41-s − 0.216·43-s + 0.944·45-s − 0.143·47-s + 3/7·49-s − 1.01·51-s + 2.50·53-s − 1.06·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(688.555\)
Root analytic conductor: \(5.12253\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7056,\ (\ :7/2, 7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(2.595243682\)
\(L(\frac12)\) \(\approx\) \(2.595243682\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_1$ \( ( 1 + p^{3} T )^{2} \)
7$C_1$ \( ( 1 - p^{3} T )^{2} \)
good5$D_{4}$ \( 1 - 264 T + 8462 p T^{2} - 264 p^{7} T^{3} + p^{14} T^{4} \)
11$D_{4}$ \( 1 + 4980 T + 38737606 T^{2} + 4980 p^{7} T^{3} + p^{14} T^{4} \)
13$D_{4}$ \( 1 + 10148 T + 75576846 T^{2} + 10148 p^{7} T^{3} + p^{14} T^{4} \)
17$D_{4}$ \( 1 - 17832 T + 345159502 T^{2} - 17832 p^{7} T^{3} + p^{14} T^{4} \)
19$D_{4}$ \( 1 - 6256 T + 1754965926 T^{2} - 6256 p^{7} T^{3} + p^{14} T^{4} \)
23$D_{4}$ \( 1 - 14052 T + 6233591566 T^{2} - 14052 p^{7} T^{3} + p^{14} T^{4} \)
29$D_{4}$ \( 1 - 243588 T + 49005121054 T^{2} - 243588 p^{7} T^{3} + p^{14} T^{4} \)
31$D_{4}$ \( 1 - 470824 T + 110401476030 T^{2} - 470824 p^{7} T^{3} + p^{14} T^{4} \)
37$D_{4}$ \( 1 - 311116 T + 134140188030 T^{2} - 311116 p^{7} T^{3} + p^{14} T^{4} \)
41$D_{4}$ \( 1 - 919248 T + 579002453902 T^{2} - 919248 p^{7} T^{3} + p^{14} T^{4} \)
43$D_{4}$ \( 1 + 112616 T + 296638211478 T^{2} + 112616 p^{7} T^{3} + p^{14} T^{4} \)
47$D_{4}$ \( 1 + 102456 T + 463813338910 T^{2} + 102456 p^{7} T^{3} + p^{14} T^{4} \)
53$D_{4}$ \( 1 - 2720028 T + 3726830950846 T^{2} - 2720028 p^{7} T^{3} + p^{14} T^{4} \)
59$D_{4}$ \( 1 + 19008 T - 1663332768746 T^{2} + 19008 p^{7} T^{3} + p^{14} T^{4} \)
61$D_{4}$ \( 1 + 925148 T + 2505443574174 T^{2} + 925148 p^{7} T^{3} + p^{14} T^{4} \)
67$D_{4}$ \( 1 + 2053424 T + 3053533142214 T^{2} + 2053424 p^{7} T^{3} + p^{14} T^{4} \)
71$D_{4}$ \( 1 + 869508 T + 14567098422382 T^{2} + 869508 p^{7} T^{3} + p^{14} T^{4} \)
73$D_{4}$ \( 1 - 3505228 T + 11289516695814 T^{2} - 3505228 p^{7} T^{3} + p^{14} T^{4} \)
79$D_{4}$ \( 1 + 6640856 T + 37612376152158 T^{2} + 6640856 p^{7} T^{3} + p^{14} T^{4} \)
83$D_{4}$ \( 1 - 7856760 T + 50010766932550 T^{2} - 7856760 p^{7} T^{3} + p^{14} T^{4} \)
89$D_{4}$ \( 1 + 9330384 T + 109971781045486 T^{2} + 9330384 p^{7} T^{3} + p^{14} T^{4} \)
97$D_{4}$ \( 1 + 2220212 T + 72228119801046 T^{2} + 2220212 p^{7} T^{3} + p^{14} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.07298236927330899799685060146, −12.41572301081568519032734679827, −11.98280634167297158135393728015, −11.69389303285291398627335024541, −10.69411363917267662296796304112, −10.52883124420500626076695136953, −9.759871594598183547888901563070, −9.729496477687285842576923119222, −8.492286254031459160380319956461, −7.957714055616341217747331968084, −7.34609027537248213398591864923, −6.65629557105576025846837552656, −5.85269113435209719579174919818, −5.53290838544673194859807209315, −4.65275517417342177839413203443, −4.54542822684898988517015776932, −2.76767638564978787478410188387, −2.42049584670003522996715729942, −1.12859982641745709776145661935, −0.67960444434266820959569190865, 0.67960444434266820959569190865, 1.12859982641745709776145661935, 2.42049584670003522996715729942, 2.76767638564978787478410188387, 4.54542822684898988517015776932, 4.65275517417342177839413203443, 5.53290838544673194859807209315, 5.85269113435209719579174919818, 6.65629557105576025846837552656, 7.34609027537248213398591864923, 7.957714055616341217747331968084, 8.492286254031459160380319956461, 9.729496477687285842576923119222, 9.759871594598183547888901563070, 10.52883124420500626076695136953, 10.69411363917267662296796304112, 11.69389303285291398627335024541, 11.98280634167297158135393728015, 12.41572301081568519032734679827, 13.07298236927330899799685060146

Graph of the $Z$-function along the critical line