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Properties

Label 2-930-155.37-c1-0-25
Degree $2$
Conductor $930$
Sign $-0.810 + 0.585i$
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 + (−0.258 + 0.965i)3-s + 1.00i·4-s + (0.871 + 2.05i)5-s + (0.866 − 0.500i)6-s + (1.08 − 4.03i)7-s + (0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (0.839 − 2.07i)10-s + (−4.05 − 2.33i)11-s + (−0.965 − 0.258i)12-s + (−0.901 + 0.241i)13-s + (−3.61 + 2.08i)14-s + (−2.21 + 0.308i)15-s − 1.00·16-s + (−4.54 − 1.21i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.149 + 0.557i)3-s + 0.500i·4-s + (0.389 + 0.920i)5-s + (0.353 − 0.204i)6-s + (0.408 − 1.52i)7-s + (0.250 − 0.250i)8-s + (−0.288 − 0.166i)9-s + (0.265 − 0.655i)10-s + (−1.22 − 0.705i)11-s + (−0.278 − 0.0747i)12-s + (−0.250 + 0.0670i)13-s + (−0.967 + 0.558i)14-s + (−0.571 + 0.0797i)15-s − 0.250·16-s + (−1.10 − 0.295i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.810 + 0.585i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.810 + 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.810 + 0.585i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.810 + 0.585i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.129443 - 0.400156i\)
\(L(\frac12)\) \(\approx\) \(0.129443 - 0.400156i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 + (-0.871 - 2.05i)T \)
31 \( 1 + (-0.275 + 5.56i)T \)
good7 \( 1 + (-1.08 + 4.03i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (4.05 + 2.33i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.901 - 0.241i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (4.54 + 1.21i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.927 - 0.535i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.172 - 0.172i)T + 23iT^{2} \)
29 \( 1 + 4.36T + 29T^{2} \)
37 \( 1 + (3.61 + 0.968i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.68 - 4.64i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.997 + 3.72i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (3.65 + 3.65i)T + 47iT^{2} \)
53 \( 1 + (-7.41 + 1.98i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (10.5 - 6.10i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 9.63iT - 61T^{2} \)
67 \( 1 + (-9.86 + 2.64i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.20 - 2.07i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.53 - 1.48i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (6.03 + 10.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-12.9 + 3.47i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 4.34T + 89T^{2} \)
97 \( 1 + (-5.74 - 5.74i)T + 97iT^{2} \)
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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

−10.06492086533592479630257705269, −9.120676869890113768491008443674, −8.003918129313852331006542762395, −7.35380151844734911238370448213, −6.44839585288093512839796805819, −5.21667886627972425234142501316, −4.14793852009407630368092439238, −3.24393710953034750303201000839, −2.10363454943381514741752475779, −0.21951225347349248264133480032, 1.75915453412027495172036216259, 2.48202277405124789846966364590, 4.71737929357397498952955905136, 5.31125018652053986873784099959, 6.02636808599834417519513558552, 7.08921056218833361016586707743, 8.073957736540965129326387551728, 8.657745355840603011361465964290, 9.260755459044199724705287843576, 10.24327553823803322243464041987

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