Properties

Label 2-930-465.464-c1-0-55
Degree $2$
Conductor $930$
Sign $0.954 + 0.299i$
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  + 2-s + (1.69 + 0.341i)3-s + 4-s + (1.23 − 1.86i)5-s + (1.69 + 0.341i)6-s − 0.843i·7-s + 8-s + (2.76 + 1.16i)9-s + (1.23 − 1.86i)10-s − 2.73·11-s + (1.69 + 0.341i)12-s + 1.61·13-s − 0.843i·14-s + (2.74 − 2.73i)15-s + 16-s + 3.82i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.980 + 0.197i)3-s + 0.5·4-s + (0.554 − 0.832i)5-s + (0.693 + 0.139i)6-s − 0.318i·7-s + 0.353·8-s + (0.922 + 0.387i)9-s + (0.392 − 0.588i)10-s − 0.824·11-s + (0.490 + 0.0987i)12-s + 0.448·13-s − 0.225i·14-s + (0.707 − 0.706i)15-s + 0.250·16-s + 0.927i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 + 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.954 + 0.299i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (929, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.954 + 0.299i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.59847 - 0.550616i\)
\(L(\frac12)\) \(\approx\) \(3.59847 - 0.550616i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-1.69 - 0.341i)T \)
5 \( 1 + (-1.23 + 1.86i)T \)
31 \( 1 + (-2.58 - 4.93i)T \)
good7 \( 1 + 0.843iT - 7T^{2} \)
11 \( 1 + 2.73T + 11T^{2} \)
13 \( 1 - 1.61T + 13T^{2} \)
17 \( 1 - 3.82iT - 17T^{2} \)
19 \( 1 + 0.779T + 19T^{2} \)
23 \( 1 + 3.42iT - 23T^{2} \)
29 \( 1 + 4.68T + 29T^{2} \)
37 \( 1 - 4.77T + 37T^{2} \)
41 \( 1 + 7.71iT - 41T^{2} \)
43 \( 1 + 6.37T + 43T^{2} \)
47 \( 1 + 12.0T + 47T^{2} \)
53 \( 1 - 1.15iT - 53T^{2} \)
59 \( 1 - 3.35iT - 59T^{2} \)
61 \( 1 - 15.3iT - 61T^{2} \)
67 \( 1 + 8.81iT - 67T^{2} \)
71 \( 1 - 13.6iT - 71T^{2} \)
73 \( 1 + 11.0T + 73T^{2} \)
79 \( 1 + 4.05iT - 79T^{2} \)
83 \( 1 + 8.96iT - 83T^{2} \)
89 \( 1 - 4.16T + 89T^{2} \)
97 \( 1 - 0.560iT - 97T^{2} \)
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

−10.24666256926741407700229649321, −9.038126367301191606742239809643, −8.408046652506675864988489443403, −7.61871209719507411100725272000, −6.50950337185104499644045297513, −5.49547362619638900318572438345, −4.59351596798787577395003539922, −3.78336748127653419461121807694, −2.60681195378565219540602375653, −1.53051592666516648154349317544, 1.85343991564110460467345537691, 2.78196333480990104638056839180, 3.47551800693550432264063667873, 4.76946544554067386851179676221, 5.83099747185864602724830789484, 6.68005606535366355286789766337, 7.53429067242062377899693945348, 8.239758457594035206880044711166, 9.493902239396030849204249095732, 9.923221704040701079031556537420

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