Properties

Label 2-930-155.37-c1-0-7
Degree $2$
Conductor $930$
Sign $-0.923 - 0.383i$
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 + (−0.786 + 2.09i)5-s + (0.866 − 0.500i)6-s + (−1.17 + 4.37i)7-s + (0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (2.03 − 0.924i)10-s + (1.73 + 1.00i)11-s + (−0.965 − 0.258i)12-s + (−0.612 + 0.164i)13-s + (3.91 − 2.26i)14-s + (−1.81 − 1.30i)15-s − 1.00·16-s + (5.31 + 1.42i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.149 + 0.557i)3-s + 0.500i·4-s + (−0.351 + 0.936i)5-s + (0.353 − 0.204i)6-s + (−0.442 + 1.65i)7-s + (0.250 − 0.250i)8-s + (−0.288 − 0.166i)9-s + (0.643 − 0.292i)10-s + (0.522 + 0.301i)11-s + (−0.278 − 0.0747i)12-s + (−0.169 + 0.0454i)13-s + (1.04 − 0.604i)14-s + (−0.469 − 0.335i)15-s − 0.250·16-s + (1.28 + 0.345i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 - 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.923 - 0.383i$
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: \((2,\ 930,\ (\ :1/2),\ -0.923 - 0.383i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.148702 + 0.746131i\)
\(L(\frac12)\) \(\approx\) \(0.148702 + 0.746131i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.786 - 2.09i)T \)
31 \( 1 + (-1.69 + 5.30i)T \)
good7 \( 1 + (1.17 - 4.37i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-1.73 - 1.00i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.612 - 0.164i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-5.31 - 1.42i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.88 - 1.08i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.15 - 6.15i)T + 23iT^{2} \)
29 \( 1 + 9.68T + 29T^{2} \)
37 \( 1 + (2.38 + 0.638i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (4.62 - 8.01i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.50 + 9.36i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (2.08 + 2.08i)T + 47iT^{2} \)
53 \( 1 + (-3.72 + 0.998i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.109 - 0.0633i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 1.06iT - 61T^{2} \)
67 \( 1 + (13.4 - 3.60i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-3.51 + 6.09i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-8.24 + 2.20i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.85 - 8.41i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.11 + 1.36i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 3.58T + 89T^{2} \)
97 \( 1 + (7.94 + 7.94i)T + 97iT^{2} \)
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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

−10.33510852225316600423031009339, −9.567224443692301520418050987495, −9.095837166232718331261763122142, −8.060872439311093239290166743977, −7.14745478964555825907501048731, −6.06917351736932968773724359756, −5.31074734065907979164978994258, −3.76874635085381173032028225571, −3.12284151676131621989297598308, −1.99541668956884845080092826207, 0.47137766406388583495971094796, 1.32766324822869858961162537060, 3.37295706643748601863307120746, 4.45905440886691353073275923451, 5.40682462442061195385093331448, 6.57253941087056051912564732234, 7.22834696759914978129130941227, 7.87139716447334751817276211956, 8.789029136948024288198982934173, 9.546992930787343240764613668830

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