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Properties

Label 2-930-155.92-c1-0-2
Degree $2$
Conductor $930$
Sign $-0.443 - 0.896i$
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  + (−0.707 − 0.707i)2-s + (0.707 + 0.707i)3-s + 1.00i·4-s + (1.06 + 1.96i)5-s − 1.00i·6-s + (−2.96 − 2.96i)7-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (0.636 − 2.14i)10-s + 3.23i·11-s + (−0.707 + 0.707i)12-s + (−1.33 − 1.33i)13-s + 4.19i·14-s + (−0.636 + 2.14i)15-s − 1.00·16-s + (−2.89 + 2.89i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.408 + 0.408i)3-s + 0.500i·4-s + (0.476 + 0.879i)5-s − 0.408i·6-s + (−1.12 − 1.12i)7-s + (0.250 − 0.250i)8-s + 0.333i·9-s + (0.201 − 0.677i)10-s + 0.975i·11-s + (−0.204 + 0.204i)12-s + (−0.371 − 0.371i)13-s + 1.12i·14-s + (−0.164 + 0.553i)15-s − 0.250·16-s + (−0.700 + 0.700i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.443 - 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.443 - 0.896i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.443 - 0.896i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.404944 + 0.651943i\)
\(L(\frac12)\) \(\approx\) \(0.404944 + 0.651943i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-1.06 - 1.96i)T \)
31 \( 1 + (3.10 - 4.62i)T \)
good7 \( 1 + (2.96 + 2.96i)T + 7iT^{2} \)
11 \( 1 - 3.23iT - 11T^{2} \)
13 \( 1 + (1.33 + 1.33i)T + 13iT^{2} \)
17 \( 1 + (2.89 - 2.89i)T - 17iT^{2} \)
19 \( 1 - 3.68iT - 19T^{2} \)
23 \( 1 + (1.35 + 1.35i)T + 23iT^{2} \)
29 \( 1 - 10.3T + 29T^{2} \)
37 \( 1 + (6.67 - 6.67i)T - 37iT^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 + (-3.51 - 3.51i)T + 43iT^{2} \)
47 \( 1 + (7.51 + 7.51i)T + 47iT^{2} \)
53 \( 1 + (-2.49 - 2.49i)T + 53iT^{2} \)
59 \( 1 + 0.694iT - 59T^{2} \)
61 \( 1 + 3.53iT - 61T^{2} \)
67 \( 1 + (6.09 + 6.09i)T + 67iT^{2} \)
71 \( 1 + 8.76T + 71T^{2} \)
73 \( 1 + (-4.41 - 4.41i)T + 73iT^{2} \)
79 \( 1 + 14.6T + 79T^{2} \)
83 \( 1 + (-8.30 - 8.30i)T + 83iT^{2} \)
89 \( 1 - 15.9T + 89T^{2} \)
97 \( 1 + (-5.33 - 5.33i)T + 97iT^{2} \)
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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

−10.22277595887804787409140136040, −9.965529235356392764764285113994, −8.858933353611190770238061139448, −7.87344395395781788335944623627, −6.90364670063441162800338635580, −6.45658593330359909229461710081, −4.80734894231216714039009187901, −3.69897492695206783497408765057, −3.02352254386678623050532788716, −1.79899151841537410707629413220, 0.37953485397035594645343736421, 2.07629914004172132335645354236, 3.07570488580402054467787266953, 4.71039583058076556045729710331, 5.71734411091883292455839958637, 6.36164520251898300073694652213, 7.20825308498112081640162194837, 8.491064324416364458143551439885, 8.884535400232721945350703840288, 9.409879710929158569111052845408

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