Properties

Label 2-403-403.25-c1-0-8
Degree $2$
Conductor $403$
Sign $0.907 - 0.419i$
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  + 0.177i·2-s + (0.311 + 0.538i)3-s + 1.96·4-s + (−1.82 − 1.05i)5-s + (−0.0956 + 0.0552i)6-s + (−1.99 + 1.15i)7-s + 0.704i·8-s + (1.30 − 2.26i)9-s + (0.187 − 0.324i)10-s + (3.83 + 2.21i)11-s + (0.612 + 1.06i)12-s + (3.46 + 1.01i)13-s + (−0.204 − 0.354i)14-s − 1.31i·15-s + 3.81·16-s + (3.63 + 6.29i)17-s + ⋯
L(s)  = 1  + 0.125i·2-s + (0.179 + 0.311i)3-s + 0.984·4-s + (−0.816 − 0.471i)5-s + (−0.0390 + 0.0225i)6-s + (−0.754 + 0.435i)7-s + 0.249i·8-s + (0.435 − 0.754i)9-s + (0.0591 − 0.102i)10-s + (1.15 + 0.668i)11-s + (0.176 + 0.306i)12-s + (0.959 + 0.280i)13-s + (−0.0546 − 0.0947i)14-s − 0.338i·15-s + 0.952·16-s + (0.881 + 1.52i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.907 - 0.419i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.907 - 0.419i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62451 + 0.357147i\)
\(L(\frac12)\) \(\approx\) \(1.62451 + 0.357147i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-3.46 - 1.01i)T \)
31 \( 1 + (4.14 + 3.71i)T \)
good2 \( 1 - 0.177iT - 2T^{2} \)
3 \( 1 + (-0.311 - 0.538i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.82 + 1.05i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.99 - 1.15i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.83 - 2.21i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.63 - 6.29i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.84 + 3.37i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 9.13T + 23T^{2} \)
29 \( 1 + 0.638T + 29T^{2} \)
37 \( 1 + (2.94 - 1.69i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.49 + 0.861i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.66 + 2.88i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.94iT - 47T^{2} \)
53 \( 1 + (5.41 - 9.38i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.57 + 0.911i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 1.62T + 61T^{2} \)
67 \( 1 + (-1.49 - 0.864i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.34 + 3.66i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.36 + 0.790i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.85 + 4.93i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.14 + 1.23i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 12.4iT - 89T^{2} \)
97 \( 1 + 7.23iT - 97T^{2} \)
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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

−11.66163176917414211461729871494, −10.30458950233275269764005566326, −9.531809036346804712783419134381, −8.582920598358955884965342889430, −7.56779500298588120055944012218, −6.55183042412259934028877522175, −5.85089773817192947632017166710, −4.01782643333826597799755665487, −3.48669076278249376782039837738, −1.56288146900282479171938159045, 1.37347144934698086727554780675, 3.19463370863733970645952017165, 3.71073999444051855932748630760, 5.66846792305457657756123379918, 6.68937520578433872182314256011, 7.45392499506192400988321137131, 8.062831330649625539012300611785, 9.611643433473904600136411306525, 10.37317599257820160836420920965, 11.44459499349086095648487497493

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