Properties

Degree 2
Conductor $ 23 \cdot 349 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.57·2-s − 1.48·3-s + 4.64·4-s − 1.39·5-s + 3.82·6-s − 5.10·7-s − 6.80·8-s − 0.799·9-s + 3.59·10-s − 1.30·11-s − 6.88·12-s + 0.398·13-s + 13.1·14-s + 2.06·15-s + 8.26·16-s + 2.72·17-s + 2.06·18-s + 0.445·19-s − 6.47·20-s + 7.57·21-s + 3.36·22-s + 23-s + 10.0·24-s − 3.05·25-s − 1.02·26-s + 5.63·27-s − 23.7·28-s + ⋯
L(s)  = 1  − 1.82·2-s − 0.856·3-s + 2.32·4-s − 0.623·5-s + 1.56·6-s − 1.93·7-s − 2.40·8-s − 0.266·9-s + 1.13·10-s − 0.394·11-s − 1.98·12-s + 0.110·13-s + 3.51·14-s + 0.534·15-s + 2.06·16-s + 0.660·17-s + 0.485·18-s + 0.102·19-s − 1.44·20-s + 1.65·21-s + 0.718·22-s + 0.208·23-s + 2.06·24-s − 0.611·25-s − 0.201·26-s + 1.08·27-s − 4.48·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8027\)    =    \(23 \cdot 349\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8027} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.001735189982$
$L(\frac12)$  $\approx$  $0.001735189982$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{23,\;349\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{23,\;349\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad23 \( 1 - T \)
349 \( 1 + T \)
good2 \( 1 + 2.57T + 2T^{2} \)
3 \( 1 + 1.48T + 3T^{2} \)
5 \( 1 + 1.39T + 5T^{2} \)
7 \( 1 + 5.10T + 7T^{2} \)
11 \( 1 + 1.30T + 11T^{2} \)
13 \( 1 - 0.398T + 13T^{2} \)
17 \( 1 - 2.72T + 17T^{2} \)
19 \( 1 - 0.445T + 19T^{2} \)
29 \( 1 + 7.89T + 29T^{2} \)
31 \( 1 + 1.20T + 31T^{2} \)
37 \( 1 - 1.57T + 37T^{2} \)
41 \( 1 - 0.346T + 41T^{2} \)
43 \( 1 - 1.60T + 43T^{2} \)
47 \( 1 - 10.7T + 47T^{2} \)
53 \( 1 + 1.77T + 53T^{2} \)
59 \( 1 + 3.20T + 59T^{2} \)
61 \( 1 + 11.0T + 61T^{2} \)
67 \( 1 + 9.27T + 67T^{2} \)
71 \( 1 + 6.33T + 71T^{2} \)
73 \( 1 - 2.18T + 73T^{2} \)
79 \( 1 + 16.0T + 79T^{2} \)
83 \( 1 + 8.96T + 83T^{2} \)
89 \( 1 + 14.8T + 89T^{2} \)
97 \( 1 - 4.83T + 97T^{2} \)
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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

−7.62668014239465558772763770583, −7.42768793292304729857343889648, −6.61539434115366631653181105797, −5.94779367150515291049331608486, −5.59648548666527064019409197934, −4.08199934459543079594419353401, −3.16931206866180412226296916553, −2.60072473718068748710875131626, −1.22256897297191426424553202428, −0.03123919810519717953580786372, 0.03123919810519717953580786372, 1.22256897297191426424553202428, 2.60072473718068748710875131626, 3.16931206866180412226296916553, 4.08199934459543079594419353401, 5.59648548666527064019409197934, 5.94779367150515291049331608486, 6.61539434115366631653181105797, 7.42768793292304729857343889648, 7.62668014239465558772763770583

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