Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $-0.476 + 0.879i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.45i·2-s + 2.80·3-s − 4.03·4-s + 3.48i·5-s − 6.87i·6-s − 3.54i·7-s + 4.99i·8-s + 4.84·9-s + 8.56·10-s − 5.60i·11-s − 11.2·12-s + (3.17 + 1.71i)13-s − 8.69·14-s + 9.76i·15-s + 4.20·16-s + 2.47·17-s + ⋯
L(s)  = 1  − 1.73i·2-s + 1.61·3-s − 2.01·4-s + 1.55i·5-s − 2.80i·6-s − 1.33i·7-s + 1.76i·8-s + 1.61·9-s + 2.70·10-s − 1.69i·11-s − 3.26·12-s + (0.879 + 0.476i)13-s − 2.32·14-s + 2.52i·15-s + 1.05·16-s + 0.601·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $-0.476 + 0.879i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (311, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 403,\ (\ :1/2),\ -0.476 + 0.879i)$
$L(1)$  $\approx$  $1.06559 - 1.78932i$
$L(\frac12)$  $\approx$  $1.06559 - 1.78932i$
$L(\frac{3}{2})$   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 \( 1 + (-3.17 - 1.71i)T \)
31 \( 1 + iT \)
good2 \( 1 + 2.45iT - 2T^{2} \)
3 \( 1 - 2.80T + 3T^{2} \)
5 \( 1 - 3.48iT - 5T^{2} \)
7 \( 1 + 3.54iT - 7T^{2} \)
11 \( 1 + 5.60iT - 11T^{2} \)
17 \( 1 - 2.47T + 17T^{2} \)
19 \( 1 - 2.94iT - 19T^{2} \)
23 \( 1 + 1.33T + 23T^{2} \)
29 \( 1 + 6.22T + 29T^{2} \)
37 \( 1 - 7.14iT - 37T^{2} \)
41 \( 1 - 4.88iT - 41T^{2} \)
43 \( 1 - 0.926T + 43T^{2} \)
47 \( 1 - 11.4iT - 47T^{2} \)
53 \( 1 + 9.27T + 53T^{2} \)
59 \( 1 + 6.89iT - 59T^{2} \)
61 \( 1 - 5.72T + 61T^{2} \)
67 \( 1 + 9.27iT - 67T^{2} \)
71 \( 1 - 0.850iT - 71T^{2} \)
73 \( 1 - 8.68iT - 73T^{2} \)
79 \( 1 + 5.69T + 79T^{2} \)
83 \( 1 - 10.1iT - 83T^{2} \)
89 \( 1 + 3.72iT - 89T^{2} \)
97 \( 1 + 11.4iT - 97T^{2} \)
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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

−11.01990760292779822110299275211, −10.16353008497116370440585672522, −9.519336064528714582837929384597, −8.381247756420331928097250050582, −7.64994976162332502406519563375, −6.33861747819215168808376642038, −3.99464358786448841446252486794, −3.45880612703748502267041251677, −2.92051329056581085359467840213, −1.47240629001336519367520001298, 2.03367276864621741948266295624, 3.93130148040072297815876640988, 4.98278445195593924207782288057, 5.74154096431866483100251360287, 7.25092210069495599112702831420, 7.961535513937547862707089435161, 8.830349181559804949183732326833, 9.001507239690158668403171343709, 9.804002962369869420935014800000, 12.11587528282559478638943092332

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