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Properties

Label 2-930-15.8-c1-0-27
Degree $2$
Conductor $930$
Sign $-0.374 - 0.927i$
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 + (1.70 + 0.292i)3-s + 1.00i·4-s + (−2.12 + 0.707i)5-s + (0.999 + 1.41i)6-s + (−0.707 + 0.707i)8-s + (2.82 + i)9-s + (−2 − 0.999i)10-s + 1.41i·11-s + (−0.292 + 1.70i)12-s + (3 + 3i)13-s + (−3.82 + 0.585i)15-s − 1.00·16-s + (−1.41 − 1.41i)17-s + (1.29 + 2.70i)18-s + 4i·19-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.985 + 0.169i)3-s + 0.500i·4-s + (−0.948 + 0.316i)5-s + (0.408 + 0.577i)6-s + (−0.250 + 0.250i)8-s + (0.942 + 0.333i)9-s + (−0.632 − 0.316i)10-s + 0.426i·11-s + (−0.0845 + 0.492i)12-s + (0.832 + 0.832i)13-s + (−0.988 + 0.151i)15-s − 0.250·16-s + (−0.342 − 0.342i)17-s + (0.304 + 0.638i)18-s + 0.917i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.374 - 0.927i$
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: \((2,\ 930,\ (\ :1/2),\ -0.374 - 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32915 + 1.96987i\)
\(L(\frac12)\) \(\approx\) \(1.32915 + 1.96987i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 - 0.292i)T \)
5 \( 1 + (2.12 - 0.707i)T \)
31 \( 1 - T \)
good7 \( 1 - 7iT^{2} \)
11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 + (-3 - 3i)T + 13iT^{2} \)
17 \( 1 + (1.41 + 1.41i)T + 17iT^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (2.82 - 2.82i)T - 23iT^{2} \)
29 \( 1 + 2.82T + 29T^{2} \)
37 \( 1 + (3 - 3i)T - 37iT^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + (-1.41 - 1.41i)T + 47iT^{2} \)
53 \( 1 + (-1.41 + 1.41i)T - 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + (-7 + 7i)T - 67iT^{2} \)
71 \( 1 + 12.7iT - 71T^{2} \)
73 \( 1 + (2 + 2i)T + 73iT^{2} \)
79 \( 1 + 2iT - 79T^{2} \)
83 \( 1 + (-4.24 + 4.24i)T - 83iT^{2} \)
89 \( 1 - 7.07T + 89T^{2} \)
97 \( 1 + (-11 + 11i)T - 97iT^{2} \)
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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

−10.26516362660806251656394043507, −9.249848223482561585402636557183, −8.506793454803538295341123847528, −7.72973692806917229822780384693, −7.11885553515330570538620492867, −6.17944588651546687327925668722, −4.77768891981928012535306282329, −3.95117509700163715706981941832, −3.35248100317936595055224117306, −1.95917235941421539842317946104, 0.888377897893704031347493990203, 2.42141718024908486135479049505, 3.49016883387830525602920195176, 4.07264686419896078577388497806, 5.17688625951214733387887437133, 6.41930049700097427792334693327, 7.38125021931317654040917094091, 8.393004231512962646892554969591, 8.710977943529454386373962051368, 9.842904749617860989148117663611

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