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Properties

Label 2-403-31.25-c1-0-28
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: $\chi_{403} (118, \cdot )$
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

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

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