Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $0.953 - 0.301i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.44i·2-s + 0.0382·3-s − 0.100·4-s + 3.85i·5-s − 0.0554i·6-s + 1.66i·7-s − 2.75i·8-s − 2.99·9-s + 5.59·10-s + 1.88i·11-s − 0.00384·12-s + (1.08 + 3.43i)13-s + 2.40·14-s + 0.147i·15-s − 4.19·16-s + 3.75·17-s + ⋯
L(s)  = 1  − 1.02i·2-s + 0.0220·3-s − 0.0503·4-s + 1.72i·5-s − 0.0226i·6-s + 0.628i·7-s − 0.973i·8-s − 0.999·9-s + 1.76·10-s + 0.566i·11-s − 0.00111·12-s + (0.301 + 0.953i)13-s + 0.643·14-s + 0.0381i·15-s − 1.04·16-s + 0.909·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.301i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 - 0.301i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $0.953 - 0.301i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (311, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 403,\ (\ :1/2),\ 0.953 - 0.301i)$
$L(1)$  $\approx$  $1.37363 + 0.211729i$
$L(\frac12)$  $\approx$  $1.37363 + 0.211729i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{13,\;31\}$, \(F_p(T)\) is a polynomial of degree 2. If $p \in \{13,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad13 \( 1 + (-1.08 - 3.43i)T \)
31 \( 1 - iT \)
good2 \( 1 + 1.44iT - 2T^{2} \)
3 \( 1 - 0.0382T + 3T^{2} \)
5 \( 1 - 3.85iT - 5T^{2} \)
7 \( 1 - 1.66iT - 7T^{2} \)
11 \( 1 - 1.88iT - 11T^{2} \)
17 \( 1 - 3.75T + 17T^{2} \)
19 \( 1 - 3.26iT - 19T^{2} \)
23 \( 1 - 5.31T + 23T^{2} \)
29 \( 1 - 3.11T + 29T^{2} \)
37 \( 1 + 11.5iT - 37T^{2} \)
41 \( 1 + 7.94iT - 41T^{2} \)
43 \( 1 + 9.24T + 43T^{2} \)
47 \( 1 - 6.97iT - 47T^{2} \)
53 \( 1 - 6.11T + 53T^{2} \)
59 \( 1 + 12.1iT - 59T^{2} \)
61 \( 1 - 1.75T + 61T^{2} \)
67 \( 1 - 7.94iT - 67T^{2} \)
71 \( 1 - 0.735iT - 71T^{2} \)
73 \( 1 - 13.5iT - 73T^{2} \)
79 \( 1 + 5.66T + 79T^{2} \)
83 \( 1 + 13.7iT - 83T^{2} \)
89 \( 1 - 10.2iT - 89T^{2} \)
97 \( 1 + 8.34iT - 97T^{2} \)
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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

−11.32807571851903265089601824551, −10.58800923155299158985465261114, −9.844955990935650629798750531087, −8.828600845345598277908040188106, −7.40094339122801876023979053539, −6.64564015609238126559716533211, −5.66531183879857994908749316707, −3.78282394472996448841528630712, −2.92608673831503691028816730563, −2.06046572575067476436516906155, 0.939047223363470004914495963438, 3.11572100545275452388798437008, 4.85194328306705359141015143869, 5.40641223564695592750581498102, 6.37559923499779255873739449418, 7.68359736815646197965724295136, 8.408090681681962062337480590180, 8.870215906143924106040739041685, 10.22179324222868267366938361501, 11.38273254614174062813345216825

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