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Properties

Label 2-403-31.8-c1-0-31
Degree $2$
Conductor $403$
Sign $-0.0238 - 0.999i$
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  + (0.700 − 2.15i)2-s + (−0.672 − 2.07i)3-s + (−2.54 − 1.84i)4-s − 3.69·5-s − 4.93·6-s + (0.888 + 0.645i)7-s + (−2.09 + 1.52i)8-s + (−1.40 + 1.02i)9-s + (−2.59 + 7.97i)10-s + (0.853 + 0.619i)11-s + (−2.11 + 6.50i)12-s + (−0.309 − 0.951i)13-s + (2.01 − 1.46i)14-s + (2.48 + 7.65i)15-s + (−0.126 − 0.390i)16-s + (0.552 − 0.401i)17-s + ⋯
L(s)  = 1  + (0.495 − 1.52i)2-s + (−0.388 − 1.19i)3-s + (−1.27 − 0.923i)4-s − 1.65·5-s − 2.01·6-s + (0.335 + 0.243i)7-s + (−0.740 + 0.538i)8-s + (−0.468 + 0.340i)9-s + (−0.819 + 2.52i)10-s + (0.257 + 0.186i)11-s + (−0.610 + 1.87i)12-s + (−0.0857 − 0.263i)13-s + (0.538 − 0.391i)14-s + (0.642 + 1.97i)15-s + (−0.0317 − 0.0976i)16-s + (0.134 − 0.0974i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0238 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0238 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.0238 - 0.999i$
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: \((2,\ 403,\ (\ :1/2),\ -0.0238 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.558168 + 0.571636i\)
\(L(\frac12)\) \(\approx\) \(0.558168 + 0.571636i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.56 + 0.0773i)T \)
good2 \( 1 + (-0.700 + 2.15i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (0.672 + 2.07i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + 3.69T + 5T^{2} \)
7 \( 1 + (-0.888 - 0.645i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (-0.853 - 0.619i)T + (3.39 + 10.4i)T^{2} \)
17 \( 1 + (-0.552 + 0.401i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.56 + 4.82i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (6.03 - 4.38i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.805 - 2.47i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 - 1.72T + 37T^{2} \)
41 \( 1 + (1.60 - 4.94i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (-2.41 + 7.43i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (0.0982 + 0.302i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-7.09 + 5.15i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.691 - 2.12i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 - 9.70T + 61T^{2} \)
67 \( 1 - 1.01T + 67T^{2} \)
71 \( 1 + (-0.475 + 0.345i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (13.5 + 9.81i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.50 + 1.81i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.03 + 3.18i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (6.91 + 5.02i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-0.391 - 0.284i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.29004257681993457127918656898, −10.04253976063438033519551037183, −8.774422919465069226151248796963, −7.63520485269444892012860035018, −7.05690879356853133861981195077, −5.42976077247940352279997438265, −4.26285084763129231144222828178, −3.32411910088437513057032780255, −1.88265871896093645802785728355, −0.47942501303645790353528607711, 3.91622812872993369377851433115, 4.07743645809652226206395175594, 5.12623405301601987784221314432, 6.15026219053357202209983617003, 7.40254757604746189011218485099, 7.954054881445530123152590038498, 8.850344625854523021839148717329, 10.14387530570730644189751261695, 11.09068576027282684805413642059, 11.85406709336364753646588717184

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