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Properties

Label 2-403-31.5-c1-0-2
Degree $2$
Conductor $403$
Sign $-0.775 + 0.631i$
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  + 0.352·2-s + (−1.65 + 2.87i)3-s − 1.87·4-s + (1.09 + 1.89i)5-s + (−0.584 + 1.01i)6-s + (−1.14 + 1.97i)7-s − 1.36·8-s + (−4.00 − 6.93i)9-s + (0.384 + 0.665i)10-s + (1.46 + 2.54i)11-s + (3.11 − 5.39i)12-s + (0.5 + 0.866i)13-s + (−0.402 + 0.696i)14-s − 7.24·15-s + 3.27·16-s + (1.85 − 3.20i)17-s + ⋯
L(s)  = 1  + 0.248·2-s + (−0.957 + 1.65i)3-s − 0.938·4-s + (0.488 + 0.845i)5-s + (−0.238 + 0.413i)6-s + (−0.431 + 0.748i)7-s − 0.482·8-s + (−1.33 − 2.31i)9-s + (0.121 + 0.210i)10-s + (0.442 + 0.766i)11-s + (0.898 − 1.55i)12-s + (0.138 + 0.240i)13-s + (−0.107 + 0.186i)14-s − 1.87·15-s + 0.817·16-s + (0.448 − 0.777i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.775 + 0.631i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.775 + 0.631i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (222, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.775 + 0.631i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.190689 - 0.535681i\)
\(L(\frac12)\) \(\approx\) \(0.190689 - 0.535681i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.56 - 0.210i)T \)
good2 \( 1 - 0.352T + 2T^{2} \)
3 \( 1 + (1.65 - 2.87i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.09 - 1.89i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.14 - 1.97i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.46 - 2.54i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.85 + 3.20i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.373 + 0.646i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.68T + 23T^{2} \)
29 \( 1 - 5.11T + 29T^{2} \)
37 \( 1 + (3.95 - 6.84i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.63 + 4.56i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.99 - 5.18i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.63T + 47T^{2} \)
53 \( 1 + (-2.20 - 3.81i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.14 - 3.70i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 3.34T + 61T^{2} \)
67 \( 1 + (-5.54 - 9.60i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.53 + 2.66i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.48 + 2.58i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.59 + 11.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.61 - 11.4i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 8.82T + 89T^{2} \)
97 \( 1 - 0.0441T + 97T^{2} \)
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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

−11.90931852030850616619131413442, −10.67368715003300939334357759482, −9.809672543176734451220961542257, −9.614710790114069900945955637525, −8.588821424292783952860502651492, −6.63656601358523235007274061107, −5.83924868235633564672379360192, −5.01272792526928641521444542687, −4.07893768226102552079140998261, −3.04003977842839335438656560979, 0.41076834277143517846926651391, 1.56081566265770574882069616143, 3.72609775467863188883572290900, 5.22211212893411152895169983625, 5.84579468902838889823472295288, 6.70145945020244065878017565189, 7.975391961845819906903398775925, 8.569678442533974821118607597904, 9.866037462179292103778670505102, 10.85742436670472427564684927093

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