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Properties

Label 2-930-155.123-c1-0-10
Degree $2$
Conductor $930$
Sign $0.732 - 0.681i$
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 + (0.975 + 2.01i)5-s − 1.00i·6-s + (−3.17 + 3.17i)7-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + (2.11 + 0.732i)10-s + 5.75i·11-s + (−0.707 − 0.707i)12-s + (0.935 − 0.935i)13-s + 4.48i·14-s + (2.11 + 0.732i)15-s − 1.00·16-s + (−2.02 − 2.02i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.408 − 0.408i)3-s − 0.500i·4-s + (0.436 + 0.899i)5-s − 0.408i·6-s + (−1.19 + 1.19i)7-s + (−0.250 − 0.250i)8-s − 0.333i·9-s + (0.668 + 0.231i)10-s + 1.73i·11-s + (−0.204 − 0.204i)12-s + (0.259 − 0.259i)13-s + 1.19i·14-s + (0.545 + 0.189i)15-s − 0.250·16-s + (−0.490 − 0.490i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 - 0.681i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.732 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.732 - 0.681i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.732 - 0.681i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84278 + 0.724572i\)
\(L(\frac12)\) \(\approx\) \(1.84278 + 0.724572i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.975 - 2.01i)T \)
31 \( 1 + (3.66 - 4.18i)T \)
good7 \( 1 + (3.17 - 3.17i)T - 7iT^{2} \)
11 \( 1 - 5.75iT - 11T^{2} \)
13 \( 1 + (-0.935 + 0.935i)T - 13iT^{2} \)
17 \( 1 + (2.02 + 2.02i)T + 17iT^{2} \)
19 \( 1 - 6.40iT - 19T^{2} \)
23 \( 1 + (-4.01 + 4.01i)T - 23iT^{2} \)
29 \( 1 - 4.54T + 29T^{2} \)
37 \( 1 + (-3.68 - 3.68i)T + 37iT^{2} \)
41 \( 1 - 0.446T + 41T^{2} \)
43 \( 1 + (-8.48 + 8.48i)T - 43iT^{2} \)
47 \( 1 + (7.10 - 7.10i)T - 47iT^{2} \)
53 \( 1 + (2.35 - 2.35i)T - 53iT^{2} \)
59 \( 1 + 1.60iT - 59T^{2} \)
61 \( 1 + 12.4iT - 61T^{2} \)
67 \( 1 + (-6.43 + 6.43i)T - 67iT^{2} \)
71 \( 1 - 3.97T + 71T^{2} \)
73 \( 1 + (3.59 - 3.59i)T - 73iT^{2} \)
79 \( 1 + 3.68T + 79T^{2} \)
83 \( 1 + (-2.02 + 2.02i)T - 83iT^{2} \)
89 \( 1 + 14.0T + 89T^{2} \)
97 \( 1 + (-13.2 + 13.2i)T - 97iT^{2} \)
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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

−9.994321155828602684567461608793, −9.615251139500428224832249808146, −8.717274746474524908388267678768, −7.39994515058588187560287948657, −6.58178994123906994829140882983, −6.02432807628144033240079716126, −4.86081977934395137449682084800, −3.47705552656946962416938747348, −2.67575525601705707378165578708, −1.93280800522691856220993385241, 0.74173936470842483351699394203, 2.85478805799776656314189484467, 3.75635698031650804230533781543, 4.53794787989769258920180044140, 5.68309358786823700146954446938, 6.41182373278636772319802752696, 7.33374801482620651655547797574, 8.434819428139972866571975320402, 9.061010738637582006969671774732, 9.731460183634559325951432564010

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