Properties

Degree 2
Conductor 2
Sign $-1$
Motivic weight 41
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.04e6·2-s + 5.04e9·3-s + 1.09e12·4-s − 4.85e13·5-s − 5.28e15·6-s − 1.19e17·7-s − 1.15e18·8-s − 1.10e19·9-s + 5.08e19·10-s + 3.15e21·11-s + 5.54e21·12-s − 1.14e22·13-s + 1.25e23·14-s − 2.44e23·15-s + 1.20e24·16-s − 2.67e25·17-s + 1.15e25·18-s + 6.79e25·19-s − 5.33e25·20-s − 6.02e26·21-s − 3.30e27·22-s − 1.35e28·23-s − 5.81e27·24-s − 4.31e28·25-s + 1.19e28·26-s − 2.39e29·27-s − 1.31e29·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.835·3-s + 1/2·4-s − 0.227·5-s − 0.590·6-s − 0.565·7-s − 0.353·8-s − 0.302·9-s + 0.160·10-s + 1.41·11-s + 0.417·12-s − 0.166·13-s + 0.399·14-s − 0.189·15-s + 1/4·16-s − 1.59·17-s + 0.213·18-s + 0.414·19-s − 0.113·20-s − 0.472·21-s − 0.999·22-s − 1.64·23-s − 0.295·24-s − 0.948·25-s + 0.117·26-s − 1.08·27-s − 0.282·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(42-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+41/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(41\)
character  :  $\chi_{2} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 2,\ (\ :41/2),\ -1)\)
\(L(21)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{43}{2})\)   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 \( 1 + p^{20} T \)
good3 \( 1 - 6918404 p^{6} T + p^{41} T^{2} \)
5 \( 1 + 1940167805226 p^{2} T + p^{41} T^{2} \)
7 \( 1 + 2436580524421432 p^{2} T + p^{41} T^{2} \)
11 \( 1 - \)\(28\!\cdots\!32\)\( p T + p^{41} T^{2} \)
13 \( 1 + 67516566191869487746 p^{2} T + p^{41} T^{2} \)
17 \( 1 + \)\(15\!\cdots\!74\)\( p T + p^{41} T^{2} \)
19 \( 1 - \)\(18\!\cdots\!60\)\( p^{2} T + p^{41} T^{2} \)
23 \( 1 + \)\(13\!\cdots\!04\)\( T + p^{41} T^{2} \)
29 \( 1 - \)\(13\!\cdots\!10\)\( T + p^{41} T^{2} \)
31 \( 1 - \)\(98\!\cdots\!52\)\( p T + p^{41} T^{2} \)
37 \( 1 + \)\(59\!\cdots\!94\)\( p T + p^{41} T^{2} \)
41 \( 1 + \)\(50\!\cdots\!38\)\( T + p^{41} T^{2} \)
43 \( 1 + \)\(31\!\cdots\!84\)\( T + p^{41} T^{2} \)
47 \( 1 - \)\(13\!\cdots\!92\)\( T + p^{41} T^{2} \)
53 \( 1 + \)\(32\!\cdots\!14\)\( T + p^{41} T^{2} \)
59 \( 1 - \)\(34\!\cdots\!20\)\( T + p^{41} T^{2} \)
61 \( 1 + \)\(97\!\cdots\!78\)\( T + p^{41} T^{2} \)
67 \( 1 - \)\(16\!\cdots\!52\)\( T + p^{41} T^{2} \)
71 \( 1 - \)\(11\!\cdots\!32\)\( T + p^{41} T^{2} \)
73 \( 1 - \)\(19\!\cdots\!66\)\( T + p^{41} T^{2} \)
79 \( 1 + \)\(56\!\cdots\!80\)\( T + p^{41} T^{2} \)
83 \( 1 + \)\(60\!\cdots\!84\)\( T + p^{41} T^{2} \)
89 \( 1 - \)\(11\!\cdots\!90\)\( T + p^{41} T^{2} \)
97 \( 1 + \)\(63\!\cdots\!98\)\( T + p^{41} T^{2} \)
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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

−17.42635804582235831545073259863, −15.65383792604942111326636438046, −13.96921270041742277130727864878, −11.72817598372064561374838745052, −9.598517821897693977014552375650, −8.377379889930806691296240666831, −6.55461737338123172920406101733, −3.67838765534465907917156277613, −2.01911258865289250076112753959, 0, 2.01911258865289250076112753959, 3.67838765534465907917156277613, 6.55461737338123172920406101733, 8.377379889930806691296240666831, 9.598517821897693977014552375650, 11.72817598372064561374838745052, 13.96921270041742277130727864878, 15.65383792604942111326636438046, 17.42635804582235831545073259863

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