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Properties

Label 2-403-31.5-c1-0-20
Degree $2$
Conductor $403$
Sign $0.902 - 0.431i$
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  + 2.38·2-s + (−0.113 + 0.196i)3-s + 3.70·4-s + (0.669 + 1.15i)5-s + (−0.270 + 0.468i)6-s + (−0.988 + 1.71i)7-s + 4.06·8-s + (1.47 + 2.55i)9-s + (1.59 + 2.76i)10-s + (−2.83 − 4.90i)11-s + (−0.419 + 0.726i)12-s + (0.5 + 0.866i)13-s + (−2.35 + 4.08i)14-s − 0.303·15-s + 2.29·16-s + (2.83 − 4.90i)17-s + ⋯
L(s)  = 1  + 1.68·2-s + (−0.0654 + 0.113i)3-s + 1.85·4-s + (0.299 + 0.518i)5-s + (−0.110 + 0.191i)6-s + (−0.373 + 0.646i)7-s + 1.43·8-s + (0.491 + 0.851i)9-s + (0.505 + 0.875i)10-s + (−0.853 − 1.47i)11-s + (−0.121 + 0.209i)12-s + (0.138 + 0.240i)13-s + (−0.630 + 1.09i)14-s − 0.0783·15-s + 0.573·16-s + (0.687 − 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 - 0.431i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 - 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.902 - 0.431i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (222, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.902 - 0.431i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.24328 + 0.735195i\)
\(L(\frac12)\) \(\approx\) \(3.24328 + 0.735195i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-5.45 - 1.10i)T \)
good2 \( 1 - 2.38T + 2T^{2} \)
3 \( 1 + (0.113 - 0.196i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.669 - 1.15i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.988 - 1.71i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.83 + 4.90i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.83 + 4.90i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.631 + 1.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 6.56T + 23T^{2} \)
29 \( 1 + 0.229T + 29T^{2} \)
37 \( 1 + (-2.08 + 3.60i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.42 + 4.20i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.75 - 4.77i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 + (-6.76 - 11.7i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.304 + 0.526i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 0.648T + 61T^{2} \)
67 \( 1 + (-4.20 - 7.28i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.35 - 5.81i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (8.36 + 14.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.77 - 9.99i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.43 + 4.21i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3.58T + 89T^{2} \)
97 \( 1 + 0.808T + 97T^{2} \)
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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

−11.53861893176009618931040455444, −10.74191655778481455132307058487, −9.839728130326812361437413346755, −8.372424685060790111363385713550, −7.24187531666481071646327297423, −6.10703510592999254103240521330, −5.55168989485861538422835145114, −4.54560824851266360043034497157, −3.15941161532603956980566545478, −2.47394002879212553547264381926, 1.79940584013441767883238996735, 3.42304835879653780224904280839, 4.28257638189223642557470542646, 5.23396239427724378462419037366, 6.24109715009815925147274644542, 7.04568414948797490761804265293, 8.123078126866201838866662549954, 9.868960009156081342062664295275, 10.21186083525671922862792609862, 11.71097016343101208499292631097

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