Properties

Label 2-930-155.129-c1-0-8
Degree $2$
Conductor $930$
Sign $0.711 - 0.702i$
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 − 0.312i)5-s + (0.5 − 0.866i)6-s + (−1.24 − 0.718i)7-s i·8-s + (0.499 + 0.866i)9-s + (0.312 − 2.21i)10-s + (1.06 + 1.84i)11-s + (0.866 + 0.5i)12-s + (−0.930 + 0.537i)13-s + (0.718 − 1.24i)14-s + (1.76 + 1.37i)15-s + 16-s + (−3.17 − 1.83i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.499 − 0.288i)3-s − 0.5·4-s + (−0.990 − 0.139i)5-s + (0.204 − 0.353i)6-s + (−0.470 − 0.271i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (0.0987 − 0.700i)10-s + (0.321 + 0.556i)11-s + (0.249 + 0.144i)12-s + (−0.258 + 0.149i)13-s + (0.191 − 0.332i)14-s + (0.454 + 0.355i)15-s + 0.250·16-s + (−0.769 − 0.444i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.711 - 0.702i$
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: \((2,\ 930,\ (\ :1/2),\ 0.711 - 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.788410 + 0.323867i\)
\(L(\frac12)\) \(\approx\) \(0.788410 + 0.323867i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.21 + 0.312i)T \)
31 \( 1 + (-5.36 + 1.50i)T \)
good7 \( 1 + (1.24 + 0.718i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.06 - 1.84i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.930 - 0.537i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.17 + 1.83i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.71 + 2.97i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.935iT - 23T^{2} \)
29 \( 1 - 3.18T + 29T^{2} \)
37 \( 1 + (-3.50 - 2.02i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.85 - 10.1i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-8.03 - 4.63i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 7.84iT - 47T^{2} \)
53 \( 1 + (-9.23 + 5.32i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.43 - 2.48i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 3.56T + 61T^{2} \)
67 \( 1 + (1.09 - 0.631i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.0395 + 0.0685i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.91 - 1.68i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.23 + 9.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (11.0 - 6.36i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 1.28T + 89T^{2} \)
97 \( 1 + 15.8iT - 97T^{2} \)
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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

−10.00643111999392914247104198634, −9.283438295088658297190235660560, −8.293320048727170410983023913833, −7.43106072962967075229787098646, −6.88378687335902049367551838055, −6.07831075932834184711911380663, −4.73356279159716856593881684309, −4.31700698795578927891606818251, −2.86225575980170641169010805860, −0.800469403778899248501810839976, 0.69122214950115523877809358621, 2.57788430385010971130856473978, 3.68888489874124673254458506250, 4.33803663791636235138554907113, 5.51924148412585607511430062902, 6.44518059097110354846372209970, 7.48575600001336057827200134735, 8.518648763872307110716505677995, 9.158843538917326314842766603051, 10.25509549536083581512178432158

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