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Properties

Label 2-930-155.37-c1-0-17
Degree $2$
Conductor $930$
Sign $0.260 - 0.965i$
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 + (−0.0918 − 2.23i)5-s + (−0.866 + 0.500i)6-s + (−0.264 + 0.988i)7-s + (−0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (1.51 − 1.64i)10-s + (3.16 + 1.82i)11-s + (−0.965 − 0.258i)12-s + (6.11 − 1.63i)13-s + (−0.886 + 0.511i)14-s + (2.18 + 0.489i)15-s − 1.00·16-s + (1.52 + 0.409i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.149 + 0.557i)3-s + 0.500i·4-s + (−0.0410 − 0.999i)5-s + (−0.353 + 0.204i)6-s + (−0.100 + 0.373i)7-s + (−0.250 + 0.250i)8-s + (−0.288 − 0.166i)9-s + (0.479 − 0.520i)10-s + (0.954 + 0.551i)11-s + (−0.278 − 0.0747i)12-s + (1.69 − 0.454i)13-s + (−0.236 + 0.136i)14-s + (0.563 + 0.126i)15-s − 0.250·16-s + (0.370 + 0.0993i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.260 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.260 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.260 - 0.965i$
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: \((2,\ 930,\ (\ :1/2),\ 0.260 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64685 + 1.26190i\)
\(L(\frac12)\) \(\approx\) \(1.64685 + 1.26190i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0918 + 2.23i)T \)
31 \( 1 + (-4.65 - 3.04i)T \)
good7 \( 1 + (0.264 - 0.988i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-3.16 - 1.82i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-6.11 + 1.63i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-1.52 - 0.409i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (5.10 - 2.94i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.163 - 0.163i)T + 23iT^{2} \)
29 \( 1 - 8.26T + 29T^{2} \)
37 \( 1 + (0.315 + 0.0844i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.76 - 3.05i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.79 - 10.4i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-4.01 - 4.01i)T + 47iT^{2} \)
53 \( 1 + (1.27 - 0.340i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (6.62 - 3.82i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 14.6iT - 61T^{2} \)
67 \( 1 + (11.5 - 3.09i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-7.89 + 13.6i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.45 + 0.925i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.18 + 8.98i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.70 - 0.725i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 - 16.0T + 89T^{2} \)
97 \( 1 + (9.77 + 9.77i)T + 97iT^{2} \)
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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

−10.18676060339469265707399315027, −9.196137421077435329600188820052, −8.547723338212927414939487522451, −7.948417884944914465446567976654, −6.30331964599325926139302414710, −6.10786823438431240856802960890, −4.82775958373344508942028329291, −4.22944441334657972377443491189, −3.24126548348454785136644087644, −1.38402443061648715132149527407, 1.02474211522947422688763777032, 2.38315398339009885589372544280, 3.52990963821400372216627455756, 4.21870305169616619755289175353, 5.77611868887415422691848118774, 6.51599041526302150446899720175, 6.92125249427858003640876300787, 8.317578956440351708961436957205, 8.997736467549896500917900909995, 10.34425449010852081829193842930

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