Properties

Label 2-930-155.129-c1-0-20
Degree $2$
Conductor $930$
Sign $0.803 - 0.595i$
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 + (1.97 − 1.04i)5-s + (−0.5 + 0.866i)6-s + (−2.10 − 1.21i)7-s i·8-s + (0.499 + 0.866i)9-s + (1.04 + 1.97i)10-s + (−0.345 − 0.598i)11-s + (−0.866 − 0.5i)12-s + (1.66 − 0.959i)13-s + (1.21 − 2.10i)14-s + (2.23 + 0.0874i)15-s + 16-s + (4.39 + 2.53i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.499 + 0.288i)3-s − 0.5·4-s + (0.884 − 0.465i)5-s + (−0.204 + 0.353i)6-s + (−0.796 − 0.460i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (0.329 + 0.625i)10-s + (−0.104 − 0.180i)11-s + (−0.249 − 0.144i)12-s + (0.460 − 0.266i)13-s + (0.325 − 0.563i)14-s + (0.576 + 0.0225i)15-s + 0.250·16-s + (1.06 + 0.616i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.803 - 0.595i$
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: \((2,\ 930,\ (\ :1/2),\ 0.803 - 0.595i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91774 + 0.632649i\)
\(L(\frac12)\) \(\approx\) \(1.91774 + 0.632649i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.97 + 1.04i)T \)
31 \( 1 + (-4.90 - 2.63i)T \)
good7 \( 1 + (2.10 + 1.21i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.345 + 0.598i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.66 + 0.959i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-4.39 - 2.53i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.14 + 3.71i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 6.84iT - 23T^{2} \)
29 \( 1 - 6.99T + 29T^{2} \)
37 \( 1 + (1.27 + 0.735i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.94 + 10.2i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-9.13 - 5.27i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.99iT - 47T^{2} \)
53 \( 1 + (-9.46 + 5.46i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.78 + 3.09i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 7.77T + 61T^{2} \)
67 \( 1 + (12.2 - 7.09i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.66 + 9.80i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.34 - 3.08i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.37 - 12.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.756 + 0.436i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 7.31T + 89T^{2} \)
97 \( 1 - 9.79iT - 97T^{2} \)
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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

−10.07199881328693871646010869974, −9.209967794024669190803050610372, −8.585217344407854335316717725868, −7.63804207178388357369447093945, −6.71796886239872257685949492184, −5.80865582995410356839998379302, −5.08450357301595354655836444917, −3.86000917894880882702432032257, −2.91815818605547732026305537769, −1.14499409644014240131132631921, 1.29190133518139796890750627879, 2.66058701003981838113631293706, 3.09880746387023927079744581924, 4.48714955281275439873821114116, 5.79550204648614551699864881125, 6.41438947238277636767381767599, 7.50891088974877702401977557509, 8.567797490280698123362803737594, 9.296817805455456666479331004512, 10.09269526328063764492630580156

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