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Properties

Label 2-403-31.2-c1-0-7
Degree $2$
Conductor $403$
Sign $0.958 + 0.284i$
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  + (−1.78 − 1.29i)2-s + (1.61 − 1.17i)3-s + (0.879 + 2.70i)4-s − 1.55·5-s − 4.38·6-s + (0.929 + 2.86i)7-s + (0.575 − 1.77i)8-s + (0.299 − 0.921i)9-s + (2.77 + 2.01i)10-s + (1.55 + 4.77i)11-s + (4.58 + 3.33i)12-s + (0.809 − 0.587i)13-s + (2.04 − 6.29i)14-s + (−2.50 + 1.82i)15-s + (1.28 − 0.936i)16-s + (0.488 − 1.50i)17-s + ⋯
L(s)  = 1  + (−1.25 − 0.914i)2-s + (0.930 − 0.676i)3-s + (0.439 + 1.35i)4-s − 0.696·5-s − 1.79·6-s + (0.351 + 1.08i)7-s + (0.203 − 0.626i)8-s + (0.0998 − 0.307i)9-s + (0.876 + 0.637i)10-s + (0.467 + 1.43i)11-s + (1.32 + 0.962i)12-s + (0.224 − 0.163i)13-s + (0.547 − 1.68i)14-s + (−0.647 + 0.470i)15-s + (0.322 − 0.234i)16-s + (0.118 − 0.364i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.284i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 + 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.958 + 0.284i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.958 + 0.284i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.861530 - 0.125170i\)
\(L(\frac12)\) \(\approx\) \(0.861530 - 0.125170i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (-5.36 - 1.50i)T \)
good2 \( 1 + (1.78 + 1.29i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-1.61 + 1.17i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + 1.55T + 5T^{2} \)
7 \( 1 + (-0.929 - 2.86i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (-1.55 - 4.77i)T + (-8.89 + 6.46i)T^{2} \)
17 \( 1 + (-0.488 + 1.50i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-4.84 - 3.51i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (0.936 - 2.88i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (6.37 + 4.63i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + 6.12T + 37T^{2} \)
41 \( 1 + (1.47 + 1.07i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-6.35 - 4.61i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-2.87 + 2.08i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.93 + 9.03i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-3.07 + 2.23i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 - 11.9T + 61T^{2} \)
67 \( 1 + 10.5T + 67T^{2} \)
71 \( 1 + (0.625 - 1.92i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (3.89 + 11.9i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (4.06 - 12.4i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-4.26 - 3.09i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-4.17 - 12.8i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (0.159 + 0.491i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.39591579434580611395398255952, −10.00175825941270638335107658525, −9.386061830406211546978824334134, −8.511559531640126137933331485398, −7.83089956826456011857260640872, −7.24668321601992198087977077273, −5.41009178112295767965890747203, −3.66003237567730610740225058094, −2.42534671820990569036504129016, −1.60863019922628763952060593826, 0.844293338998172137345564605957, 3.36696235561867480540612245756, 4.14398325219903517109343018265, 5.86095289413404917579003631776, 7.08402345002022719111473821112, 7.78325128878550943862451803522, 8.650318613443250711726012841437, 9.070870525995489366741649224723, 10.14809879414719051876366787263, 10.87810162343957831330686462333

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