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Properties

Label 2-930-15.2-c1-0-50
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: $\chi_{930} (497, \cdot )$
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

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

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