↑

Properties

Label 2-930-15.2-c1-0-28
Degree $2$
Conductor $930$
Sign $0.287 - 0.957i$
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 + (1.22 + 1.22i)3-s − 1.00i·4-s + (1.67 − 1.48i)5-s − 1.73·6-s + (2 + 2i)7-s + (0.707 + 0.707i)8-s + 2.99i·9-s + (−0.133 + 2.23i)10-s + 2.31i·11-s + (1.22 − 1.22i)12-s + (3.36 − 3.36i)13-s − 2.82·14-s + (3.86 + 0.232i)15-s − 1.00·16-s + (−0.707 + 0.707i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.707 + 0.707i)3-s − 0.500i·4-s + (0.748 − 0.663i)5-s − 0.707·6-s + (0.755 + 0.755i)7-s + (0.250 + 0.250i)8-s + 0.999i·9-s + (−0.0423 + 0.705i)10-s + 0.696i·11-s + (0.353 − 0.353i)12-s + (0.933 − 0.933i)13-s − 0.755·14-s + (0.998 + 0.0599i)15-s − 0.250·16-s + (−0.171 + 0.171i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.287 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.287 - 0.957i$
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: \((2,\ 930,\ (\ :1/2),\ 0.287 - 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58721 + 1.18054i\)
\(L(\frac12)\) \(\approx\) \(1.58721 + 1.18054i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.22 - 1.22i)T \)
5 \( 1 + (-1.67 + 1.48i)T \)
31 \( 1 + T \)
good7 \( 1 + (-2 - 2i)T + 7iT^{2} \)
11 \( 1 - 2.31iT - 11T^{2} \)
13 \( 1 + (-3.36 + 3.36i)T - 13iT^{2} \)
17 \( 1 + (0.707 - 0.707i)T - 17iT^{2} \)
19 \( 1 + 3iT - 19T^{2} \)
23 \( 1 + (-2.44 - 2.44i)T + 23iT^{2} \)
29 \( 1 + 4.62T + 29T^{2} \)
37 \( 1 + (0.464 + 0.464i)T + 37iT^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 + (-3.46 + 3.46i)T - 43iT^{2} \)
47 \( 1 + (-0.189 + 0.189i)T - 47iT^{2} \)
53 \( 1 + (4.89 + 4.89i)T + 53iT^{2} \)
59 \( 1 - 9.52T + 59T^{2} \)
61 \( 1 - 10.1T + 61T^{2} \)
67 \( 1 + (-0.169 - 0.169i)T + 67iT^{2} \)
71 \( 1 - 15.3iT - 71T^{2} \)
73 \( 1 + (9.46 - 9.46i)T - 73iT^{2} \)
79 \( 1 + 3.73iT - 79T^{2} \)
83 \( 1 + (4.43 + 4.43i)T + 83iT^{2} \)
89 \( 1 + 9.89T + 89T^{2} \)
97 \( 1 + (8.36 + 8.36i)T + 97iT^{2} \)
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

−9.918844242847229885783329926381, −9.295618855461325293476775572418, −8.559718968362633775860130277272, −8.138309154734410307495692150739, −6.99176791643592202353709559694, −5.57583282502306350281295510506, −5.25800839134963510428848778882, −4.14084141125305610164538372945, −2.60110090098288527662325285209, −1.53794715201500429327008880443, 1.21354039676173014338116008607, 2.07947240219472408664006697898, 3.25654876085376603150565709073, 4.14975675347117337125165469115, 5.81376017620633419163026734156, 6.73627062811984027178115679687, 7.43398011297721304321587641552, 8.340828891248057473784715476948, 9.023896574271439578266607618873, 9.810719322616079483460295573999

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