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Properties

Label 2-930-155.123-c1-0-30
Degree $2$
Conductor $930$
Sign $-0.999 + 0.00560i$
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 + (−1.87 − 1.21i)5-s − 1.00i·6-s + (−0.853 + 0.853i)7-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + (−2.18 + 0.469i)10-s − 2.87i·11-s + (−0.707 − 0.707i)12-s + (0.276 − 0.276i)13-s + 1.20i·14-s + (−2.18 + 0.469i)15-s − 1.00·16-s + (−3.42 − 3.42i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.408 − 0.408i)3-s − 0.500i·4-s + (−0.839 − 0.542i)5-s − 0.408i·6-s + (−0.322 + 0.322i)7-s + (−0.250 − 0.250i)8-s − 0.333i·9-s + (−0.691 + 0.148i)10-s − 0.866i·11-s + (−0.204 − 0.204i)12-s + (0.0767 − 0.0767i)13-s + 0.322i·14-s + (−0.564 + 0.121i)15-s − 0.250·16-s + (−0.831 − 0.831i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00560i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.999 + 0.00560i$
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: \((2,\ 930,\ (\ :1/2),\ -0.999 + 0.00560i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00357586 - 1.27611i\)
\(L(\frac12)\) \(\approx\) \(0.00357586 - 1.27611i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.87 + 1.21i)T \)
31 \( 1 + (5.04 + 2.36i)T \)
good7 \( 1 + (0.853 - 0.853i)T - 7iT^{2} \)
11 \( 1 + 2.87iT - 11T^{2} \)
13 \( 1 + (-0.276 + 0.276i)T - 13iT^{2} \)
17 \( 1 + (3.42 + 3.42i)T + 17iT^{2} \)
19 \( 1 - 2.20iT - 19T^{2} \)
23 \( 1 + (-0.351 + 0.351i)T - 23iT^{2} \)
29 \( 1 + 8.45T + 29T^{2} \)
37 \( 1 + (3.34 + 3.34i)T + 37iT^{2} \)
41 \( 1 - 4.86T + 41T^{2} \)
43 \( 1 + (-4.42 + 4.42i)T - 43iT^{2} \)
47 \( 1 + (1.22 - 1.22i)T - 47iT^{2} \)
53 \( 1 + (-1.78 + 1.78i)T - 53iT^{2} \)
59 \( 1 + 9.30iT - 59T^{2} \)
61 \( 1 + 1.65iT - 61T^{2} \)
67 \( 1 + (6.27 - 6.27i)T - 67iT^{2} \)
71 \( 1 - 12.7T + 71T^{2} \)
73 \( 1 + (-6.49 + 6.49i)T - 73iT^{2} \)
79 \( 1 - 5.71T + 79T^{2} \)
83 \( 1 + (-8.55 + 8.55i)T - 83iT^{2} \)
89 \( 1 - 4.78T + 89T^{2} \)
97 \( 1 + (-8.62 + 8.62i)T - 97iT^{2} \)
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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

−9.353021539780390114439035353345, −9.018524903880189150577134517559, −7.975523069994716820085870670158, −7.18570149867442956669572356383, −6.05239316079699421535332906712, −5.18728032314792562458853406201, −3.99397137743161577008494241770, −3.29219865394652918879896234905, −2.06048885606443003125076854213, −0.46229690628825828727098766348, 2.28379724956224207591744733669, 3.55605192645622193016785462354, 4.11062446655388535105225866846, 5.07222354238296885860733000209, 6.36260972108889532649278634035, 7.14610725619297877414051570672, 7.75032169287534593558571610565, 8.741048492461908960211288998583, 9.532546056930057119303755717727, 10.63158823231444228831223518680

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