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Properties

Label 2-930-155.129-c1-0-19
Degree $2$
Conductor $930$
Sign $-0.0675 - 0.997i$
Analytic cond. $7.42608$
Root an. cond. $2.72508$
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  + i·2-s + (0.866 + 0.5i)3-s − 4-s + (2.22 + 0.168i)5-s + (−0.5 + 0.866i)6-s + (3.61 + 2.08i)7-s − i·8-s + (0.499 + 0.866i)9-s + (−0.168 + 2.22i)10-s + (−0.324 − 0.561i)11-s + (−0.866 − 0.5i)12-s + (−1.99 + 1.15i)13-s + (−2.08 + 3.61i)14-s + (1.84 + 1.26i)15-s + 16-s + (−0.894 − 0.516i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.499 + 0.288i)3-s − 0.5·4-s + (0.997 + 0.0754i)5-s + (−0.204 + 0.353i)6-s + (1.36 + 0.789i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.0533 + 0.705i)10-s + (−0.0977 − 0.169i)11-s + (−0.249 − 0.144i)12-s + (−0.553 + 0.319i)13-s + (−0.558 + 0.967i)14-s + (0.476 + 0.325i)15-s + 0.250·16-s + (−0.217 − 0.125i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0675 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0675 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.0675 - 0.997i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.0675 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60785 + 1.72035i\)
\(L(\frac12)\) \(\approx\) \(1.60785 + 1.72035i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (-2.22 - 0.168i)T \)
31 \( 1 + (-2.68 + 4.87i)T \)
good7 \( 1 + (-3.61 - 2.08i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.324 + 0.561i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.99 - 1.15i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.894 + 0.516i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.12 - 3.68i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.34iT - 23T^{2} \)
29 \( 1 - 9.24T + 29T^{2} \)
37 \( 1 + (9.55 + 5.51i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.04 - 3.54i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.09 + 2.94i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.42iT - 47T^{2} \)
53 \( 1 + (5.52 - 3.19i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.574 + 0.995i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 8.11T + 61T^{2} \)
67 \( 1 + (-9.53 + 5.50i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.71 + 2.96i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.38 - 4.26i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.47 - 14.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.43 - 1.40i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 - 0.0488iT - 97T^{2} \)
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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

−10.13363657414076159054881291833, −9.253411870865569728298326842931, −8.511317714793278045041794020816, −8.026183260076024198462859086183, −6.82525056717207202345259370273, −5.92983865948853049235605287013, −5.05243615684740018634240501238, −4.39899491055212986593323656542, −2.71524823087320714634631293384, −1.75898350766082363624308286392, 1.21331917534156097515880965585, 2.08277985244630019918563099023, 3.17688027714199406852425099201, 4.64456011533769417522498442186, 5.06276864252258428584922120858, 6.52516097938500785727597921004, 7.43069860154078181813784920074, 8.392319454740913110025139166257, 8.961710489872007451514308111701, 10.17995204766830470262872501311

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