Properties

Degree 2
Conductor 2
Sign $-1$
Motivic weight 39
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.24e5·2-s − 7.35e8·3-s + 2.74e11·4-s − 1.62e13·5-s − 3.85e14·6-s + 1.60e16·7-s + 1.44e17·8-s − 3.51e18·9-s − 8.50e18·10-s − 1.67e20·11-s − 2.02e20·12-s − 1.32e21·13-s + 8.41e21·14-s + 1.19e22·15-s + 7.55e22·16-s − 4.96e23·17-s − 1.84e24·18-s − 1.14e25·19-s − 4.46e24·20-s − 1.18e25·21-s − 8.77e25·22-s − 6.66e26·23-s − 1.05e26·24-s − 1.55e27·25-s − 6.93e26·26-s + 5.56e27·27-s + 4.41e27·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.365·3-s + 1/2·4-s − 0.380·5-s − 0.258·6-s + 0.532·7-s + 0.353·8-s − 0.866·9-s − 0.269·10-s − 0.825·11-s − 0.182·12-s − 0.251·13-s + 0.376·14-s + 0.138·15-s + 1/4·16-s − 0.503·17-s − 0.612·18-s − 1.33·19-s − 0.190·20-s − 0.194·21-s − 0.583·22-s − 1.86·23-s − 0.129·24-s − 0.855·25-s − 0.177·26-s + 0.681·27-s + 0.266·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(40-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+39/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(39\)
character  :  $\chi_{2} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 2,\ (\ :39/2),\ -1)\)
\(L(20)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{41}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 2$,\(F_p(T)\) is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - p^{19} T \)
good3 \( 1 + 9079732 p^{4} T + p^{39} T^{2} \)
5 \( 1 + 129809431866 p^{3} T + p^{39} T^{2} \)
7 \( 1 - 2292866539412552 p T + p^{39} T^{2} \)
11 \( 1 + 15220886148183602868 p T + p^{39} T^{2} \)
13 \( 1 + \)\(10\!\cdots\!34\)\( p T + p^{39} T^{2} \)
17 \( 1 + \)\(17\!\cdots\!94\)\( p^{2} T + p^{39} T^{2} \)
19 \( 1 + \)\(60\!\cdots\!20\)\( p T + p^{39} T^{2} \)
23 \( 1 + \)\(66\!\cdots\!72\)\( T + p^{39} T^{2} \)
29 \( 1 - \)\(15\!\cdots\!50\)\( p T + p^{39} T^{2} \)
31 \( 1 - \)\(15\!\cdots\!92\)\( T + p^{39} T^{2} \)
37 \( 1 + \)\(69\!\cdots\!58\)\( p T + p^{39} T^{2} \)
41 \( 1 - \)\(51\!\cdots\!42\)\( T + p^{39} T^{2} \)
43 \( 1 - \)\(78\!\cdots\!28\)\( T + p^{39} T^{2} \)
47 \( 1 - \)\(24\!\cdots\!04\)\( T + p^{39} T^{2} \)
53 \( 1 - \)\(69\!\cdots\!98\)\( T + p^{39} T^{2} \)
59 \( 1 + \)\(20\!\cdots\!00\)\( T + p^{39} T^{2} \)
61 \( 1 + \)\(12\!\cdots\!18\)\( T + p^{39} T^{2} \)
67 \( 1 - \)\(45\!\cdots\!64\)\( T + p^{39} T^{2} \)
71 \( 1 + \)\(99\!\cdots\!28\)\( T + p^{39} T^{2} \)
73 \( 1 - \)\(81\!\cdots\!18\)\( T + p^{39} T^{2} \)
79 \( 1 + \)\(85\!\cdots\!40\)\( T + p^{39} T^{2} \)
83 \( 1 - \)\(71\!\cdots\!48\)\( T + p^{39} T^{2} \)
89 \( 1 + \)\(13\!\cdots\!90\)\( T + p^{39} T^{2} \)
97 \( 1 - \)\(76\!\cdots\!34\)\( T + p^{39} T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.58342460668778859415805672800, −15.76001905867721474706666976285, −14.12477162382757872031745634333, −12.21831936355779020696548705989, −10.80569525993874776600701524501, −8.063512381102993696726985795027, −5.98118294901305269554181450553, −4.39472302074607937542041430925, −2.38168870150251759706220308002, 0, 2.38168870150251759706220308002, 4.39472302074607937542041430925, 5.98118294901305269554181450553, 8.063512381102993696726985795027, 10.80569525993874776600701524501, 12.21831936355779020696548705989, 14.12477162382757872031745634333, 15.76001905867721474706666976285, 17.58342460668778859415805672800

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