↑

Properties

Label 2-105-105.59-c1-0-11
Degree $2$
Conductor $105$
Sign $-0.967 + 0.251i$
Analytic cond. $0.838429$
Root an. cond. $0.915657$
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  + (−1.25 − 2.17i)2-s + (1.51 − 0.831i)3-s + (−2.14 + 3.71i)4-s + (−1.93 − 1.12i)5-s + (−3.71 − 2.25i)6-s + (−1.65 − 2.06i)7-s + 5.74·8-s + (1.61 − 2.52i)9-s + (−0.0219 + 5.60i)10-s + (1.48 + 0.859i)11-s + (−0.172 + 7.42i)12-s + 0.360·13-s + (−2.39 + 6.18i)14-s + (−3.87 − 0.104i)15-s + (−2.91 − 5.04i)16-s + (2.20 + 1.27i)17-s + ⋯
L(s)  = 1  + (−0.886 − 1.53i)2-s + (0.877 − 0.479i)3-s + (−1.07 + 1.85i)4-s + (−0.864 − 0.503i)5-s + (−1.51 − 0.922i)6-s + (−0.627 − 0.778i)7-s + 2.03·8-s + (0.539 − 0.841i)9-s + (−0.00693 + 1.77i)10-s + (0.448 + 0.259i)11-s + (−0.0496 + 2.14i)12-s + 0.100·13-s + (−0.639 + 1.65i)14-s + (−0.999 − 0.0270i)15-s + (−0.727 − 1.26i)16-s + (0.534 + 0.308i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.251i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 + 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.967 + 0.251i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ -0.967 + 0.251i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0886035 - 0.693135i\)
\(L(\frac12)\) \(\approx\) \(0.0886035 - 0.693135i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.51 + 0.831i)T \)
5 \( 1 + (1.93 + 1.12i)T \)
7 \( 1 + (1.65 + 2.06i)T \)
good2 \( 1 + (1.25 + 2.17i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (-1.48 - 0.859i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.360T + 13T^{2} \)
17 \( 1 + (-2.20 - 1.27i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.93 + 2.84i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.25 - 2.17i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.76iT - 29T^{2} \)
31 \( 1 + (2.41 + 1.39i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.86 - 1.65i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.63T + 41T^{2} \)
43 \( 1 - 10.0iT - 43T^{2} \)
47 \( 1 + (-5.04 + 2.91i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.727 - 1.25i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.42 - 5.93i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.38 + 0.801i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.15 - 1.24i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.1iT - 71T^{2} \)
73 \( 1 + (-5.82 + 10.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.93 - 12.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.50iT - 83T^{2} \)
89 \( 1 + (6.10 + 10.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.18T + 97T^{2} \)
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

−12.95910492390945039567148113854, −12.17050193644184313534369716839, −11.26147473961433443838771385092, −9.881084074029764529338509211618, −9.201150028390897272384320128951, −8.091071454899077054615698065688, −7.20238841972164916283857988940, −4.06944875152304716455273713685, −3.14282614799782695830351145886, −1.11242623144715815535877920721, 3.42776319988933848721576775419, 5.32463863211554925339033494412, 6.78611075408351213390136176314, 7.73740564090651319238421062467, 8.700058755994300883159820047877, 9.467643526549033312584759113857, 10.54858372546152881825577098415, 12.20012620855748523887777829328, 13.93273615291246549903469243496, 14.60548427525263835066847144063

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