Properties

Label 2-403-1.1-c1-0-14
Degree $2$
Conductor $403$
Sign $1$
Analytic cond. $3.21797$
Root an. cond. $1.79387$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.299·2-s + 3.01·3-s − 1.91·4-s + 3.67·5-s − 0.903·6-s − 0.783·7-s + 1.17·8-s + 6.10·9-s − 1.10·10-s − 1.47·11-s − 5.76·12-s − 13-s + 0.234·14-s + 11.0·15-s + 3.46·16-s − 5.71·17-s − 1.82·18-s − 1.55·19-s − 7.02·20-s − 2.36·21-s + 0.442·22-s + 4.12·23-s + 3.53·24-s + 8.51·25-s + 0.299·26-s + 9.37·27-s + 1.49·28-s + ⋯
L(s)  = 1  − 0.211·2-s + 1.74·3-s − 0.955·4-s + 1.64·5-s − 0.369·6-s − 0.296·7-s + 0.414·8-s + 2.03·9-s − 0.348·10-s − 0.445·11-s − 1.66·12-s − 0.277·13-s + 0.0626·14-s + 2.86·15-s + 0.867·16-s − 1.38·17-s − 0.431·18-s − 0.356·19-s − 1.57·20-s − 0.515·21-s + 0.0943·22-s + 0.860·23-s + 0.721·24-s + 1.70·25-s + 0.0587·26-s + 1.80·27-s + 0.282·28-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(2.125573570\)
\(L(\frac12)\) \(\approx\) \(2.125573570\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + T \)
31 \( 1 - T \)
good2 \( 1 + 0.299T + 2T^{2} \)
3 \( 1 - 3.01T + 3T^{2} \)
5 \( 1 - 3.67T + 5T^{2} \)
7 \( 1 + 0.783T + 7T^{2} \)
11 \( 1 + 1.47T + 11T^{2} \)
17 \( 1 + 5.71T + 17T^{2} \)
19 \( 1 + 1.55T + 19T^{2} \)
23 \( 1 - 4.12T + 23T^{2} \)
29 \( 1 + 10.0T + 29T^{2} \)
37 \( 1 - 5.12T + 37T^{2} \)
41 \( 1 - 6.37T + 41T^{2} \)
43 \( 1 + 1.42T + 43T^{2} \)
47 \( 1 + 11.9T + 47T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 - 7.62T + 59T^{2} \)
61 \( 1 + 5.19T + 61T^{2} \)
67 \( 1 + 2.69T + 67T^{2} \)
71 \( 1 + 5.68T + 71T^{2} \)
73 \( 1 - 9.03T + 73T^{2} \)
79 \( 1 + 3.98T + 79T^{2} \)
83 \( 1 + 0.997T + 83T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + 6.88T + 97T^{2} \)
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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

−10.79972994508412708630134820078, −9.827051263715422338893085395217, −9.351692807441014053677075641791, −8.840146597104196439587687571165, −7.87192162869237479144096174485, −6.70847664759941524155761044549, −5.34591239953407421839771722474, −4.18690033350946735162270303109, −2.82601134736842073380180582416, −1.84577727613477550226818928670, 1.84577727613477550226818928670, 2.82601134736842073380180582416, 4.18690033350946735162270303109, 5.34591239953407421839771722474, 6.70847664759941524155761044549, 7.87192162869237479144096174485, 8.840146597104196439587687571165, 9.351692807441014053677075641791, 9.827051263715422338893085395217, 10.79972994508412708630134820078

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