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Properties

Label 2-403-31.8-c1-0-24
Degree $2$
Conductor $403$
Sign $-0.663 + 0.747i$
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  + (0.655 − 2.01i)2-s + (−0.156 − 0.481i)3-s + (−2.01 − 1.46i)4-s + 2.06·5-s − 1.07·6-s + (0.413 + 0.300i)7-s + (−0.846 + 0.615i)8-s + (2.21 − 1.61i)9-s + (1.34 − 4.15i)10-s + (0.700 + 0.509i)11-s + (−0.390 + 1.20i)12-s + (−0.309 − 0.951i)13-s + (0.876 − 0.637i)14-s + (−0.322 − 0.993i)15-s + (−0.855 − 2.63i)16-s + (−4.59 + 3.34i)17-s + ⋯
L(s)  = 1  + (0.463 − 1.42i)2-s + (−0.0904 − 0.278i)3-s + (−1.00 − 0.732i)4-s + 0.921·5-s − 0.438·6-s + (0.156 + 0.113i)7-s + (−0.299 + 0.217i)8-s + (0.739 − 0.537i)9-s + (0.426 − 1.31i)10-s + (0.211 + 0.153i)11-s + (−0.112 + 0.346i)12-s + (−0.0857 − 0.263i)13-s + (0.234 − 0.170i)14-s + (−0.0833 − 0.256i)15-s + (−0.213 − 0.658i)16-s + (−1.11 + 0.810i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.663 + 0.747i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.663 + 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.663 + 0.747i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.663 + 0.747i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.816606 - 1.81710i\)
\(L(\frac12)\) \(\approx\) \(0.816606 - 1.81710i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (-4.02 - 3.85i)T \)
good2 \( 1 + (-0.655 + 2.01i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (0.156 + 0.481i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 - 2.06T + 5T^{2} \)
7 \( 1 + (-0.413 - 0.300i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (-0.700 - 0.509i)T + (3.39 + 10.4i)T^{2} \)
17 \( 1 + (4.59 - 3.34i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.539 - 1.65i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (0.708 - 0.514i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.0954 - 0.293i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 - 2.91T + 37T^{2} \)
41 \( 1 + (-0.105 + 0.323i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (2.88 - 8.89i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (-0.355 - 1.09i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.55 + 2.58i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.554 + 1.70i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 - 9.14T + 61T^{2} \)
67 \( 1 + 6.98T + 67T^{2} \)
71 \( 1 + (0.0423 - 0.0307i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-11.6 - 8.46i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (7.26 - 5.27i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (2.28 - 7.03i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (9.14 + 6.64i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-4.98 - 3.62i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.04255393151924777471144517024, −10.05040634139440879235871323507, −9.665891157233282069216159952343, −8.436187447710854665337164013385, −6.96727714433180454211424706441, −6.01855269540287156786290256113, −4.72707912856554704531136574128, −3.74281061050438444967917940809, −2.32297493347992260273119493869, −1.39314615156246735911445096085, 2.13330725092417603082344452018, 4.22504975116542606034387922282, 4.94202874682345409416303631759, 5.94103311502131839042828821003, 6.78921151922764675097795065909, 7.57786999572110044155819591528, 8.717666886197516541319345057009, 9.602549175546890649130954526125, 10.56776628088977836404601547565, 11.55268556153606264339956093229

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