↑

Properties

Label 2-930-155.149-c1-0-16
Degree $2$
Conductor $930$
Sign $0.510 - 0.859i$
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  + i·2-s + (0.866 − 0.5i)3-s − 4-s + (0.836 + 2.07i)5-s + (0.5 + 0.866i)6-s + (1.24 − 0.718i)7-s − i·8-s + (0.499 − 0.866i)9-s + (−2.07 + 0.836i)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 + (0.374 + 0.927i)5-s + (0.204 + 0.353i)6-s + (0.470 − 0.271i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.655 + 0.264i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.510 - 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.510 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.510 - 0.859i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.510 - 0.859i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82840 + 1.04049i\)
\(L(\frac12)\) \(\approx\) \(1.82840 + 1.04049i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.836 - 2.07i)T \)
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 \( 1 + 15.8iT - 97T^{2} \)
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

−10.01361243529948757076702843417, −9.343961454449354087107529058265, −8.252691758963197276905079955081, −7.72015329614308652583705172291, −6.79373722722302669504515567952, −6.13789776290538565876845066392, −5.11683271756079701394742985813, −3.80117851863934705465836140684, −2.92800414818540455433441296263, −1.37781015142063868759981918690, 1.19357472842210078055867269376, 2.26439228933162904474882635831, 3.50267186058988463489728072106, 4.60971589803211964102314704705, 5.16958087085033664098631285787, 6.37043687502227630392062282336, 7.81155136637975554570775320587, 8.443483002302309273860963583312, 9.208927461041101790283120084832, 9.860638495705532474656869866982

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