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Properties

Label 2-930-155.149-c1-0-3
Degree $2$
Conductor $930$
Sign $0.528 - 0.849i$
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 + (0.866 − 0.5i)3-s − 4-s + (−2.21 + 0.270i)5-s + (−0.5 − 0.866i)6-s + (−0.765 + 0.442i)7-s + i·8-s + (0.499 − 0.866i)9-s + (0.270 + 2.21i)10-s + (−1.13 + 1.96i)11-s + (−0.866 + 0.5i)12-s + (−3.52 − 2.03i)13-s + (0.442 + 0.765i)14-s + (−1.78 + 1.34i)15-s + 16-s + (1.77 − 1.02i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.499 − 0.288i)3-s − 0.5·4-s + (−0.992 + 0.121i)5-s + (−0.204 − 0.353i)6-s + (−0.289 + 0.167i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (0.0856 + 0.701i)10-s + (−0.342 + 0.592i)11-s + (−0.249 + 0.144i)12-s + (−0.978 − 0.565i)13-s + (0.118 + 0.204i)14-s + (−0.461 + 0.347i)15-s + 0.250·16-s + (0.430 − 0.248i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.528 - 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.528 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.528 - 0.849i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.528 - 0.849i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.661751 + 0.367665i\)
\(L(\frac12)\) \(\approx\) \(0.661751 + 0.367665i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (2.21 - 0.270i)T \)
31 \( 1 + (0.177 - 5.56i)T \)
good7 \( 1 + (0.765 - 0.442i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.13 - 1.96i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.52 + 2.03i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.77 + 1.02i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.64 - 6.30i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.05iT - 23T^{2} \)
29 \( 1 + 4.78T + 29T^{2} \)
37 \( 1 + (2.12 - 1.22i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.43 - 5.94i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.69 + 2.71i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 10.1iT - 47T^{2} \)
53 \( 1 + (5.20 + 3.00i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.14 + 12.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 11.2T + 61T^{2} \)
67 \( 1 + (-9.40 - 5.43i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.51 - 13.0i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.86 + 1.65i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.18 + 14.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.08 + 4.08i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + 14.8iT - 97T^{2} \)
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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

−9.923381308659971287910155920303, −9.712855599658982565879889339469, −8.470982839676333893689221663095, −7.64583785835677809195088673387, −7.26896498333786217146972980236, −5.69833152651780965777215334048, −4.73197582737193079774342707722, −3.51870745066788457722515102233, −2.97359267115050260146046655001, −1.52059260389983993521008725021, 0.34191577691075407864753394066, 2.65321271639338592308774870512, 3.73748287778139584584513134420, 4.59670034774137141071695307091, 5.45783668348079765929249221653, 6.83866193050556268119921092674, 7.37181436139912313851997253297, 8.228866553579399057981117531836, 8.942577141245574541449697802921, 9.694693870309792077278264469322

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