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Properties

Label 2-403-13.3-c1-0-23
Degree $2$
Conductor $403$
Sign $0.493 - 0.869i$
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  + (1.11 + 1.92i)2-s + (−0.337 − 0.584i)3-s + (−1.47 + 2.55i)4-s + 1.70·5-s + (0.751 − 1.30i)6-s + (2.45 − 4.24i)7-s − 2.12·8-s + (1.27 − 2.20i)9-s + (1.89 + 3.27i)10-s + (1.43 + 2.48i)11-s + 1.99·12-s + (−3.48 − 0.929i)13-s + 10.9·14-s + (−0.574 − 0.994i)15-s + (0.589 + 1.02i)16-s + (−1.82 + 3.16i)17-s + ⋯
L(s)  = 1  + (0.786 + 1.36i)2-s + (−0.194 − 0.337i)3-s + (−0.738 + 1.27i)4-s + 0.760·5-s + (0.306 − 0.531i)6-s + (0.927 − 1.60i)7-s − 0.751·8-s + (0.424 − 0.734i)9-s + (0.598 + 1.03i)10-s + (0.432 + 0.749i)11-s + 0.575·12-s + (−0.966 − 0.257i)13-s + 2.91·14-s + (−0.148 − 0.256i)15-s + (0.147 + 0.255i)16-s + (−0.443 + 0.767i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.493 - 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.493 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.493 - 0.869i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (94, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.493 - 0.869i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97785 + 1.15186i\)
\(L(\frac12)\) \(\approx\) \(1.97785 + 1.15186i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (3.48 + 0.929i)T \)
31 \( 1 + T \)
good2 \( 1 + (-1.11 - 1.92i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.337 + 0.584i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 1.70T + 5T^{2} \)
7 \( 1 + (-2.45 + 4.24i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.43 - 2.48i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.82 - 3.16i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.68 - 2.91i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.08 + 1.88i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.61 + 7.99i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 + (-4.66 - 8.08i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.86 - 6.68i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.18 - 10.7i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 10.9T + 47T^{2} \)
53 \( 1 - 6.20T + 53T^{2} \)
59 \( 1 + (4.08 - 7.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.07 + 5.32i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.24 + 3.89i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.99 + 3.45i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 6.02T + 73T^{2} \)
79 \( 1 - 2.87T + 79T^{2} \)
83 \( 1 + 2.83T + 83T^{2} \)
89 \( 1 + (4.83 + 8.37i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.81 + 6.60i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.62158311375946106811544270776, −10.33156689696691496012592931261, −9.673197155712161336396445525046, −8.036905231424080261239251293225, −7.48392324957353326206225888308, −6.61587781778556682760829702129, −5.92042833757461433061067980898, −4.51552903909702675544966487975, −4.11732207300432411122278321201, −1.61989435054506108005350335980, 1.95516666497524004404112349615, 2.47714884113523268947804575017, 4.13389017426074057140137169394, 5.34001433869746491591703211331, 5.46945388547776127859468714679, 7.34092247366987972969531603445, 8.903973382859162403617706092080, 9.422284758092643405996131520757, 10.57898020397771598035777971134, 11.23255896024084775316308088606

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