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Properties

Label 2-403-13.12-c1-0-15
Degree $2$
Conductor $403$
Sign $-0.476 - 0.879i$
Analytic cond. $3.21797$
Root an. cond. $1.79387$
Motivic weight $1$
Arithmetic yes
Rational no
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

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.476 - 0.879i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.476 - 0.879i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06559 + 1.78932i\)
\(L(\frac12)\) \(\approx\) \(1.06559 + 1.78932i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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