↑

Properties

Label 2-930-15.2-c1-0-59
Degree $2$
Conductor $930$
Sign $-0.586 - 0.810i$
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  + (0.707 − 0.707i)2-s + (−0.333 − 1.69i)3-s − 1.00i·4-s + (−1.37 − 1.76i)5-s + (−1.43 − 0.965i)6-s + (−2.07 − 2.07i)7-s + (−0.707 − 0.707i)8-s + (−2.77 + 1.13i)9-s + (−2.21 − 0.270i)10-s + 3.16i·11-s + (−1.69 + 0.333i)12-s + (4.31 − 4.31i)13-s − 2.93·14-s + (−2.53 + 2.93i)15-s − 1.00·16-s + (−1.60 + 1.60i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.192 − 0.981i)3-s − 0.500i·4-s + (−0.616 − 0.787i)5-s + (−0.587 − 0.394i)6-s + (−0.783 − 0.783i)7-s + (−0.250 − 0.250i)8-s + (−0.925 + 0.378i)9-s + (−0.701 − 0.0855i)10-s + 0.955i·11-s + (−0.490 + 0.0963i)12-s + (1.19 − 1.19i)13-s − 0.783·14-s + (−0.653 + 0.756i)15-s − 0.250·16-s + (−0.389 + 0.389i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.586 - 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.586 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.586 - 0.810i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.586 - 0.810i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.436083 + 0.853782i\)
\(L(\frac12)\) \(\approx\) \(0.436083 + 0.853782i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (0.333 + 1.69i)T \)
5 \( 1 + (1.37 + 1.76i)T \)
31 \( 1 + T \)
good7 \( 1 + (2.07 + 2.07i)T + 7iT^{2} \)
11 \( 1 - 3.16iT - 11T^{2} \)
13 \( 1 + (-4.31 + 4.31i)T - 13iT^{2} \)
17 \( 1 + (1.60 - 1.60i)T - 17iT^{2} \)
19 \( 1 + 4.20iT - 19T^{2} \)
23 \( 1 + (-1.74 - 1.74i)T + 23iT^{2} \)
29 \( 1 + 6.90T + 29T^{2} \)
37 \( 1 + (-5.96 - 5.96i)T + 37iT^{2} \)
41 \( 1 + 9.79iT - 41T^{2} \)
43 \( 1 + (7.01 - 7.01i)T - 43iT^{2} \)
47 \( 1 + (-3.86 + 3.86i)T - 47iT^{2} \)
53 \( 1 + (-5.48 - 5.48i)T + 53iT^{2} \)
59 \( 1 - 1.65T + 59T^{2} \)
61 \( 1 + 5.62T + 61T^{2} \)
67 \( 1 + (11.1 + 11.1i)T + 67iT^{2} \)
71 \( 1 + 6.44iT - 71T^{2} \)
73 \( 1 + (-11.3 + 11.3i)T - 73iT^{2} \)
79 \( 1 - 11.1iT - 79T^{2} \)
83 \( 1 + (5.80 + 5.80i)T + 83iT^{2} \)
89 \( 1 + 12.5T + 89T^{2} \)
97 \( 1 + (6.14 + 6.14i)T + 97iT^{2} \)
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

−9.545206421901212380970422193648, −8.653668741390925135051547174153, −7.68988805799614323733645772530, −6.98424826363286952688421889506, −6.01967188419569603049002741173, −5.08315776105889319069324778195, −3.99889754299657241880526396180, −3.07491205584154490215761506783, −1.54240096499391327622448959688, −0.39621623192067037439668267463, 2.71544743267287295519002343010, 3.62927438255621764222888270876, 4.17926878641020489246320476524, 5.63025570414024931650321356676, 6.14433232167123158086822881847, 6.93002816042086935524207932543, 8.244637819815904753611588104640, 8.900011039633119708038212499773, 9.660068666052036684858859181727, 10.83319131901231263747748467733

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