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Properties

Label 2-403-31.25-c1-0-27
Degree $2$
Conductor $403$
Sign $-0.212 + 0.977i$
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  + 1.21·2-s + (−0.671 − 1.16i)3-s − 0.522·4-s + (1.76 − 3.06i)5-s + (−0.815 − 1.41i)6-s + (0.724 + 1.25i)7-s − 3.06·8-s + (0.599 − 1.03i)9-s + (2.15 − 3.72i)10-s + (−0.932 + 1.61i)11-s + (0.350 + 0.607i)12-s + (0.5 − 0.866i)13-s + (0.880 + 1.52i)14-s − 4.74·15-s − 2.68·16-s + (−2.73 − 4.73i)17-s + ⋯
L(s)  = 1  + 0.859·2-s + (−0.387 − 0.671i)3-s − 0.261·4-s + (0.791 − 1.37i)5-s + (−0.333 − 0.576i)6-s + (0.273 + 0.474i)7-s − 1.08·8-s + (0.199 − 0.345i)9-s + (0.680 − 1.17i)10-s + (−0.281 + 0.486i)11-s + (0.101 + 0.175i)12-s + (0.138 − 0.240i)13-s + (0.235 + 0.407i)14-s − 1.22·15-s − 0.670·16-s + (−0.662 − 1.14i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.212 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.212 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.212 + 0.977i$
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: \((2,\ 403,\ (\ :1/2),\ -0.212 + 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07379 - 1.33200i\)
\(L(\frac12)\) \(\approx\) \(1.07379 - 1.33200i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.33 - 5.05i)T \)
good2 \( 1 - 1.21T + 2T^{2} \)
3 \( 1 + (0.671 + 1.16i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.76 + 3.06i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.724 - 1.25i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.932 - 1.61i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.73 + 4.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.70 + 2.95i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.02T + 23T^{2} \)
29 \( 1 - 5.96T + 29T^{2} \)
37 \( 1 + (-5.42 - 9.39i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.08 + 1.87i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.54 - 2.67i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.89T + 47T^{2} \)
53 \( 1 + (2.36 - 4.09i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.44 + 7.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 6.70T + 61T^{2} \)
67 \( 1 + (3.73 - 6.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.70 - 8.14i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.08 - 3.61i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.84 - 8.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.30 + 9.18i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 2.23T + 89T^{2} \)
97 \( 1 + 13.6T + 97T^{2} \)
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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

−11.50478369598931487254211579534, −9.897528999024242687139159610249, −9.094625743236189665128446073513, −8.459079265876287207696725257278, −6.91507692007936263674384692957, −5.97508916635681046768524186874, −5.02954963839473785315010948624, −4.54272283035975382877228059766, −2.62155739368218434268925055762, −0.956404561328240956471754984318, 2.41059854538546251861025019965, 3.74913575801686254470023614848, 4.55765503763391016201798973928, 5.81062915805043671506974467399, 6.31729448241292877675204641663, 7.65564088120730784130908218496, 8.963997099065229908273670121801, 10.06091063509092560253590582936, 10.68467629362252145365552476618, 11.22169898235161444703155280997

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