Properties

Label 4-7098e2-1.1-c1e2-0-10
Degree $4$
Conductor $50381604$
Sign $1$
Analytic cond. $3212.37$
Root an. cond. $7.52846$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·3-s + 3·4-s − 4·6-s + 2·7-s − 4·8-s + 3·9-s + 8·11-s + 6·12-s − 4·14-s + 5·16-s + 6·17-s − 6·18-s + 8·19-s + 4·21-s − 16·22-s + 8·23-s − 8·24-s + 2·25-s + 4·27-s + 6·28-s + 16·29-s − 6·31-s − 6·32-s + 16·33-s − 12·34-s + 9·36-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.63·6-s + 0.755·7-s − 1.41·8-s + 9-s + 2.41·11-s + 1.73·12-s − 1.06·14-s + 5/4·16-s + 1.45·17-s − 1.41·18-s + 1.83·19-s + 0.872·21-s − 3.41·22-s + 1.66·23-s − 1.63·24-s + 2/5·25-s + 0.769·27-s + 1.13·28-s + 2.97·29-s − 1.07·31-s − 1.06·32-s + 2.78·33-s − 2.05·34-s + 3/2·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(50381604\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(3212.37\)
Root analytic conductor: \(7.52846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7098} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 50381604,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.880887207\)
\(L(\frac12)\) \(\approx\) \(5.880887207\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
13 \( 1 \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 - 6 T + 31 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 8 T + 59 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 + 6 T + 59 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 2 T + 75 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 8 T + 98 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 20 T + 203 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 16 T + 179 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 95 T^{2} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 4 T + 111 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 10 T + 119 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 8 T + 114 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 163 T^{2} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 20 T + 275 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.236713336419315473911198335842, −7.908296408879606614790903769136, −7.32138610246219232948122183523, −7.18438395766331645159489783737, −7.10678371512682751386119663084, −6.56293275455036711522047823934, −6.11674307219569224778317790563, −5.93175064712730447794545039555, −5.16332621935084075352028237542, −4.99576293044204927356687923117, −4.40312208888020143210666591993, −4.16771682654335727415482345369, −3.34378688986018763777195252395, −3.26818403360503602650113425933, −3.00869747050779622771990597275, −2.50994665509960108158258842014, −1.67526975740573446117946764824, −1.42065742635199413001210269792, −1.09756481357866018475059221339, −0.859833862515387912114742919619, 0.859833862515387912114742919619, 1.09756481357866018475059221339, 1.42065742635199413001210269792, 1.67526975740573446117946764824, 2.50994665509960108158258842014, 3.00869747050779622771990597275, 3.26818403360503602650113425933, 3.34378688986018763777195252395, 4.16771682654335727415482345369, 4.40312208888020143210666591993, 4.99576293044204927356687923117, 5.16332621935084075352028237542, 5.93175064712730447794545039555, 6.11674307219569224778317790563, 6.56293275455036711522047823934, 7.10678371512682751386119663084, 7.18438395766331645159489783737, 7.32138610246219232948122183523, 7.908296408879606614790903769136, 8.236713336419315473911198335842

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