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Properties

Label 2-930-155.129-c1-0-26
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: $\chi_{930} (439, \cdot )$
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

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

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