Properties

Label 4-2646e2-1.1-c1e2-0-33
Degree $4$
Conductor $7001316$
Sign $1$
Analytic cond. $446.409$
Root an. cond. $4.59656$
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 + 3·4-s + 6·5-s + 4·8-s + 12·10-s + 2·11-s + 5·16-s + 6·19-s + 18·20-s + 4·22-s − 2·23-s + 19·25-s − 8·29-s + 6·31-s + 6·32-s + 2·37-s + 12·38-s + 24·40-s + 18·41-s + 4·43-s + 6·44-s − 4·46-s + 12·47-s + 38·50-s − 16·53-s + 12·55-s − 16·58-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 2.68·5-s + 1.41·8-s + 3.79·10-s + 0.603·11-s + 5/4·16-s + 1.37·19-s + 4.02·20-s + 0.852·22-s − 0.417·23-s + 19/5·25-s − 1.48·29-s + 1.07·31-s + 1.06·32-s + 0.328·37-s + 1.94·38-s + 3.79·40-s + 2.81·41-s + 0.609·43-s + 0.904·44-s − 0.589·46-s + 1.75·47-s + 5.37·50-s − 2.19·53-s + 1.61·55-s − 2.10·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7001316\)    =    \(2^{2} \cdot 3^{6} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(446.409\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2646} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7001316,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(15.40139003\)
\(L(\frac12)\) \(\approx\) \(15.40139003\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 6 T + 39 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 2 T + 29 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 8 T + 56 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 6 T + 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 2 T + 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 18 T + 155 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 24 T + 258 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 12 T + 152 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 6 T + 133 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 12 T + 174 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
79$C_4$ \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 12 T + 104 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 18 T + 227 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \)
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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

−9.166842514238877803478348594218, −9.148108987249869161880067523254, −7.998977716037011430582748528918, −7.893026270784226733293819080731, −7.32362188931187267450667805184, −7.05506180091244839040163674767, −6.36530676770611223618227281589, −6.13130481376390654462139224632, −5.95603870663587484067373273039, −5.73674732587861549767015390052, −5.06783775217587556021535986966, −5.02334085606076570924153620318, −4.29170669624063899082897280535, −3.99234981595643646862144211892, −3.31025776586033247714555297481, −2.85363617238687408820917285198, −2.33848688169552428754112225922, −2.17291440614683413871730989990, −1.29590909086098774447378790802, −1.19632729917276239148482596958, 1.19632729917276239148482596958, 1.29590909086098774447378790802, 2.17291440614683413871730989990, 2.33848688169552428754112225922, 2.85363617238687408820917285198, 3.31025776586033247714555297481, 3.99234981595643646862144211892, 4.29170669624063899082897280535, 5.02334085606076570924153620318, 5.06783775217587556021535986966, 5.73674732587861549767015390052, 5.95603870663587484067373273039, 6.13130481376390654462139224632, 6.36530676770611223618227281589, 7.05506180091244839040163674767, 7.32362188931187267450667805184, 7.893026270784226733293819080731, 7.998977716037011430582748528918, 9.148108987249869161880067523254, 9.166842514238877803478348594218

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