Properties

Label 4-5610e2-1.1-c1e2-0-6
Degree $4$
Conductor $31472100$
Sign $1$
Analytic cond. $2006.68$
Root an. cond. $6.69298$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 3·4-s − 2·5-s + 4·6-s − 7-s − 4·8-s + 3·9-s + 4·10-s + 2·11-s − 6·12-s + 3·13-s + 2·14-s + 4·15-s + 5·16-s + 2·17-s − 6·18-s − 3·19-s − 6·20-s + 2·21-s − 4·22-s − 11·23-s + 8·24-s + 3·25-s − 6·26-s − 4·27-s − 3·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.894·5-s + 1.63·6-s − 0.377·7-s − 1.41·8-s + 9-s + 1.26·10-s + 0.603·11-s − 1.73·12-s + 0.832·13-s + 0.534·14-s + 1.03·15-s + 5/4·16-s + 0.485·17-s − 1.41·18-s − 0.688·19-s − 1.34·20-s + 0.436·21-s − 0.852·22-s − 2.29·23-s + 1.63·24-s + 3/5·25-s − 1.17·26-s − 0.769·27-s − 0.566·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31472100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31472100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(31472100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(2006.68\)
Root analytic conductor: \(6.69298\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 31472100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
17$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 + T + 4 T^{2} + p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 3 T + 18 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 3 T + 30 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 11 T + 66 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 + 5 T + 58 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 15 T + 120 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 6 T + 62 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 14 T + 126 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 9 T + 132 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 23 T + 256 T^{2} + 23 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 - 2 T + 118 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 13 T + 198 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 - 13 T + 226 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
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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

−8.104243685636532251792514330328, −7.47307031441355263443564334187, −7.33562528050378533342065265028, −7.16039264948505804938371183058, −6.28038856076610573744188918781, −6.24348485355516752071027589089, −5.99576220393345575935860708775, −5.97191629739982875032885631960, −4.93989601376155683507650612338, −4.77125753737462418299036720647, −4.19691229532904581822426403601, −3.94856926225853955817665329859, −3.28641305453383894085972477394, −3.23995042862187386407040841262, −2.29941554869577573253683079765, −1.93280507341639125328988477365, −1.29454529277273062550006619805, −0.956544196337253874366883013927, 0, 0, 0.956544196337253874366883013927, 1.29454529277273062550006619805, 1.93280507341639125328988477365, 2.29941554869577573253683079765, 3.23995042862187386407040841262, 3.28641305453383894085972477394, 3.94856926225853955817665329859, 4.19691229532904581822426403601, 4.77125753737462418299036720647, 4.93989601376155683507650612338, 5.97191629739982875032885631960, 5.99576220393345575935860708775, 6.24348485355516752071027589089, 6.28038856076610573744188918781, 7.16039264948505804938371183058, 7.33562528050378533342065265028, 7.47307031441355263443564334187, 8.104243685636532251792514330328

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