Properties

Label 2-930-155.39-c1-0-29
Degree $2$
Conductor $930$
Sign $-0.928 + 0.371i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.951 − 0.309i)2-s + (0.951 − 0.309i)3-s + (0.809 + 0.587i)4-s + (0.618 − 2.14i)5-s − 0.999·6-s + (−1.53 + 2.11i)7-s + (−0.587 − 0.809i)8-s + (0.809 − 0.587i)9-s + (−1.25 + 1.85i)10-s + (−1.18 − 0.858i)11-s + (0.951 + 0.309i)12-s + (−4.52 + 1.47i)13-s + (2.11 − 1.53i)14-s + (−0.0762 − 2.23i)15-s + (0.309 + 0.951i)16-s + (−2.98 − 4.10i)17-s + ⋯
L(s)  = 1  + (−0.672 − 0.218i)2-s + (0.549 − 0.178i)3-s + (0.404 + 0.293i)4-s + (0.276 − 0.961i)5-s − 0.408·6-s + (−0.579 + 0.797i)7-s + (−0.207 − 0.286i)8-s + (0.269 − 0.195i)9-s + (−0.395 + 0.585i)10-s + (−0.356 − 0.258i)11-s + (0.274 + 0.0892i)12-s + (−1.25 + 0.407i)13-s + (0.564 − 0.409i)14-s + (−0.0196 − 0.577i)15-s + (0.0772 + 0.237i)16-s + (−0.723 − 0.996i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 + 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.928 + 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.928 + 0.371i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.928 + 0.371i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.121862 - 0.632006i\)
\(L(\frac12)\) \(\approx\) \(0.121862 - 0.632006i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 + 0.309i)T \)
3 \( 1 + (-0.951 + 0.309i)T \)
5 \( 1 + (-0.618 + 2.14i)T \)
31 \( 1 + (1.69 + 5.30i)T \)
good7 \( 1 + (1.53 - 2.11i)T + (-2.16 - 6.65i)T^{2} \)
11 \( 1 + (1.18 + 0.858i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (4.52 - 1.47i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.98 + 4.10i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.574 + 1.76i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (2.59 + 3.57i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.334 + 1.02i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 + 4.67iT - 37T^{2} \)
41 \( 1 + (2.57 - 7.91i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (4.48 + 1.45i)T + (34.7 + 25.2i)T^{2} \)
47 \( 1 + (-9.18 + 2.98i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.866 - 1.19i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.33 + 4.11i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + 0.894T + 61T^{2} \)
67 \( 1 + 2.19iT - 67T^{2} \)
71 \( 1 + (-9.22 + 6.70i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (7.37 - 10.1i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.95 - 2.14i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.39 + 0.453i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (1.40 + 1.02i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-1.18 + 1.63i)T + (-29.9 - 92.2i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.427342912498617421966955086582, −9.074554756185239127987843638504, −8.237621191278977450133193485189, −7.36950074336637380336406617712, −6.40548025557561125464469373853, −5.31595860822646867399595921468, −4.30428356944180470303220610931, −2.74367824580650502776351097815, −2.12825597716647160904246350066, −0.31957444014168064019347893436, 1.90004651199025899753489948192, 2.97116577673576099258524486822, 3.95500307043214715003785725835, 5.35397491377427363110204009184, 6.48003517566252614330835849571, 7.22255614017155260837383551790, 7.75736298174154258448845675252, 8.822095439608755986748383333305, 9.808752475151942473175069247622, 10.26939962434581873700015950765

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