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Properties

Label 2-930-93.26-c1-0-26
Degree $2$
Conductor $930$
Sign $-0.390 + 0.920i$
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  − i·2-s + (−1.5 − 0.866i)3-s − 4-s + (−0.866 + 0.5i)5-s + (−0.866 + 1.5i)6-s + (2 − 3.46i)7-s + i·8-s + (1.5 + 2.59i)9-s + (0.5 + 0.866i)10-s + (2.59 + 4.5i)11-s + (1.5 + 0.866i)12-s + (−3.46 − 2i)14-s + 1.73·15-s + 16-s + (2.59 − 4.5i)17-s + (2.59 − 1.5i)18-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.866 − 0.499i)3-s − 0.5·4-s + (−0.387 + 0.223i)5-s + (−0.353 + 0.612i)6-s + (0.755 − 1.30i)7-s + 0.353i·8-s + (0.5 + 0.866i)9-s + (0.158 + 0.273i)10-s + (0.783 + 1.35i)11-s + (0.433 + 0.249i)12-s + (−0.925 − 0.534i)14-s + 0.447·15-s + 0.250·16-s + (0.630 − 1.09i)17-s + (0.612 − 0.353i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.390 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.390 + 0.920i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.390 + 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.642970 - 0.971105i\)
\(L(\frac12)\) \(\approx\) \(0.642970 - 0.971105i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (1.5 + 0.866i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + (-2 - 5.19i)T \)
good7 \( 1 + (-2 + 3.46i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.59 - 4.5i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.59 + 4.5i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 5.19T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
37 \( 1 + (4.5 + 2.59i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.19 + 3i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.5 + 4.33i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 9iT - 47T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (10.3 + 6i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 13.8iT - 61T^{2} \)
67 \( 1 + (3.5 + 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.19 + 3i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-12 + 6.92i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.5 + 0.866i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 + 4T + 97T^{2} \)
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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

−10.06178923419606985866359258947, −9.229575664306202792215917049268, −7.82061252698721390639565556009, −7.27452304113327261292297110276, −6.63595746566033341382607318293, −4.92889131350663383143320303175, −4.69179122198578913378321690766, −3.43579055931173407819118864271, −1.80936429355201335079613872650, −0.78757118907454973014975984036, 1.19487658969905343734895072287, 3.31704772930903909036161592609, 4.32829470791041273020123434879, 5.38912911507031353540836683683, 5.82105856294525961704063579047, 6.65982992034449909591070772219, 7.980696058799198902254933384665, 8.631984986158468657075857793260, 9.253339176738766241973992276418, 10.33995013785011286211357560993

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