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Properties

Label 2-930-15.8-c1-0-17
Degree $2$
Conductor $930$
Sign $-0.287 - 0.957i$
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 + (−1.22 + 1.22i)3-s + 1.00i·4-s + (−1.67 − 1.48i)5-s − 1.73·6-s + (2 − 2i)7-s + (−0.707 + 0.707i)8-s − 2.99i·9-s + (−0.133 − 2.23i)10-s + 2.31i·11-s + (−1.22 − 1.22i)12-s + (3.36 + 3.36i)13-s + 2.82·14-s + (3.86 − 0.232i)15-s − 1.00·16-s + (0.707 + 0.707i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.707 + 0.707i)3-s + 0.500i·4-s + (−0.748 − 0.663i)5-s − 0.707·6-s + (0.755 − 0.755i)7-s + (−0.250 + 0.250i)8-s − 0.999i·9-s + (−0.0423 − 0.705i)10-s + 0.696i·11-s + (−0.353 − 0.353i)12-s + (0.933 + 0.933i)13-s + 0.755·14-s + (0.998 − 0.0599i)15-s − 0.250·16-s + (0.171 + 0.171i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.287 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.287 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.287 - 0.957i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.287 - 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.853387 + 1.14735i\)
\(L(\frac12)\) \(\approx\) \(0.853387 + 1.14735i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 + (1.67 + 1.48i)T \)
31 \( 1 + T \)
good7 \( 1 + (-2 + 2i)T - 7iT^{2} \)
11 \( 1 - 2.31iT - 11T^{2} \)
13 \( 1 + (-3.36 - 3.36i)T + 13iT^{2} \)
17 \( 1 + (-0.707 - 0.707i)T + 17iT^{2} \)
19 \( 1 - 3iT - 19T^{2} \)
23 \( 1 + (2.44 - 2.44i)T - 23iT^{2} \)
29 \( 1 - 4.62T + 29T^{2} \)
37 \( 1 + (0.464 - 0.464i)T - 37iT^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 + (-3.46 - 3.46i)T + 43iT^{2} \)
47 \( 1 + (0.189 + 0.189i)T + 47iT^{2} \)
53 \( 1 + (-4.89 + 4.89i)T - 53iT^{2} \)
59 \( 1 + 9.52T + 59T^{2} \)
61 \( 1 - 10.1T + 61T^{2} \)
67 \( 1 + (-0.169 + 0.169i)T - 67iT^{2} \)
71 \( 1 - 15.3iT - 71T^{2} \)
73 \( 1 + (9.46 + 9.46i)T + 73iT^{2} \)
79 \( 1 - 3.73iT - 79T^{2} \)
83 \( 1 + (-4.43 + 4.43i)T - 83iT^{2} \)
89 \( 1 - 9.89T + 89T^{2} \)
97 \( 1 + (8.36 - 8.36i)T - 97iT^{2} \)
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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

−10.42747532415520914202180092407, −9.461738144233052413286967570995, −8.511023664525162642170930159707, −7.74718294999356887006070955732, −6.82790142882314853450229932134, −5.86801080953964305215446447160, −4.83664861619730481216003305208, −4.26859777001398957237995381179, −3.64455370033652090238233428316, −1.33166042201689597615221631131, 0.71928242065889306004602204468, 2.26784404680484281523051113999, 3.27039748438159743064705793936, 4.53048892476070476856285478261, 5.54737298603282246088812831280, 6.16442336788873120291603505288, 7.18395613462812528950545993981, 8.135840924210273844994242296350, 8.737984608450268102865921147003, 10.39262426519076976382156728085

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