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Properties

Label 2-930-155.27-c1-0-5
Degree $2$
Conductor $930$
Sign $-0.678 - 0.734i$
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  + (−0.453 + 0.891i)2-s + (−0.891 + 0.453i)3-s + (−0.587 − 0.809i)4-s + (−2.23 − 0.133i)5-s − 1.00i·6-s + (0.703 + 0.111i)7-s + (0.987 − 0.156i)8-s + (0.587 − 0.809i)9-s + (1.13 − 1.92i)10-s + (0.929 + 1.27i)11-s + (0.891 + 0.453i)12-s + (3.32 − 1.69i)13-s + (−0.418 + 0.575i)14-s + (2.04 − 0.894i)15-s + (−0.309 + 0.951i)16-s + (−3.22 + 0.510i)17-s + ⋯
L(s)  = 1  + (−0.321 + 0.630i)2-s + (−0.514 + 0.262i)3-s + (−0.293 − 0.404i)4-s + (−0.998 − 0.0596i)5-s − 0.408i·6-s + (0.265 + 0.0420i)7-s + (0.349 − 0.0553i)8-s + (0.195 − 0.269i)9-s + (0.358 − 0.609i)10-s + (0.280 + 0.385i)11-s + (0.257 + 0.131i)12-s + (0.921 − 0.469i)13-s + (−0.111 + 0.153i)14-s + (0.529 − 0.230i)15-s + (−0.0772 + 0.237i)16-s + (−0.782 + 0.123i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.678 - 0.734i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.678 - 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.274224 + 0.626160i\)
\(L(\frac12)\) \(\approx\) \(0.274224 + 0.626160i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.453 - 0.891i)T \)
3 \( 1 + (0.891 - 0.453i)T \)
5 \( 1 + (2.23 + 0.133i)T \)
31 \( 1 + (-4.09 - 3.76i)T \)
good7 \( 1 + (-0.703 - 0.111i)T + (6.65 + 2.16i)T^{2} \)
11 \( 1 + (-0.929 - 1.27i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-3.32 + 1.69i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (3.22 - 0.510i)T + (16.1 - 5.25i)T^{2} \)
19 \( 1 + (-1.25 + 0.407i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (-0.137 - 0.867i)T + (-21.8 + 7.10i)T^{2} \)
29 \( 1 + (0.493 + 1.51i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + (6.52 - 6.52i)T - 37iT^{2} \)
41 \( 1 + (-1.32 - 4.07i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (2.78 - 5.46i)T + (-25.2 - 34.7i)T^{2} \)
47 \( 1 + (10.6 - 5.43i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-1.97 - 12.4i)T + (-50.4 + 16.3i)T^{2} \)
59 \( 1 + (-7.22 - 2.34i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + 7.00iT - 61T^{2} \)
67 \( 1 + (-6.16 - 6.16i)T + 67iT^{2} \)
71 \( 1 + (7.30 + 5.30i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.66 - 0.738i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-4.68 - 3.40i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.07 + 2.10i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 + (10.4 - 7.56i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (5.75 + 0.912i)T + (92.2 + 29.9i)T^{2} \)
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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

−10.38721444138528518650796934158, −9.460384942346472246966683538585, −8.490847987670274121128191596797, −7.979302445798599048702364516595, −6.90315311202344750813354190693, −6.27420939063451320303754889440, −5.06258356364653057632293230160, −4.39865586456732381128587059091, −3.28816189732074425379791932198, −1.22100213063565003384657506125, 0.45166602033931247400448660039, 1.87342596178406761427692944981, 3.41244016412957677921569674478, 4.18731743577594072376700013184, 5.23319497358156150045743337766, 6.52346362474207832022609247192, 7.22479752280696613955642028892, 8.334743092131471421635652281848, 8.728379453838852705921145659729, 9.910349160959966946376669068859

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