↑

Properties

Label 2-29-29.12-c8-0-15
Degree $2$
Conductor $29$
Sign $0.100 - 0.994i$
Analytic cond. $11.8139$
Root an. cond. $3.43714$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−15.3 − 15.3i)2-s + (5.11 + 5.11i)3-s + 218. i·4-s − 625. i·5-s − 157. i·6-s − 974.·7-s + (−583. + 583. i)8-s − 6.50e3i·9-s + (−9.62e3 + 9.62e3i)10-s + (−1.58e4 − 1.58e4i)11-s + (−1.11e3 + 1.11e3i)12-s + 3.70e4i·13-s + (1.50e4 + 1.50e4i)14-s + (3.19e3 − 3.19e3i)15-s + 7.38e4·16-s + (6.23e4 + 6.23e4i)17-s + ⋯
L(s)  = 1  + (−0.962 − 0.962i)2-s + (0.0630 + 0.0630i)3-s + 0.851i·4-s − 1.00i·5-s − 0.121i·6-s − 0.406·7-s + (−0.142 + 0.142i)8-s − 0.992i·9-s + (−0.962 + 0.962i)10-s + (−1.08 − 1.08i)11-s + (−0.0537 + 0.0537i)12-s + 1.29i·13-s + (0.390 + 0.390i)14-s + (0.0631 − 0.0631i)15-s + 1.12·16-s + (0.746 + 0.746i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.100 - 0.994i$
Analytic conductor: \(11.8139\)
Root analytic conductor: \(3.43714\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :4),\ 0.100 - 0.994i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.0501874 + 0.0453912i\)
\(L(\frac12)\) \(\approx\) \(0.0501874 + 0.0453912i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (2.02e5 - 6.77e5i)T \)
good2 \( 1 + (15.3 + 15.3i)T + 256iT^{2} \)
3 \( 1 + (-5.11 - 5.11i)T + 6.56e3iT^{2} \)
5 \( 1 + 625. iT - 3.90e5T^{2} \)
7 \( 1 + 974.T + 5.76e6T^{2} \)
11 \( 1 + (1.58e4 + 1.58e4i)T + 2.14e8iT^{2} \)
13 \( 1 - 3.70e4iT - 8.15e8T^{2} \)
17 \( 1 + (-6.23e4 - 6.23e4i)T + 6.97e9iT^{2} \)
19 \( 1 + (-1.03e5 - 1.03e5i)T + 1.69e10iT^{2} \)
23 \( 1 + 4.82e5T + 7.83e10T^{2} \)
31 \( 1 + (4.22e5 + 4.22e5i)T + 8.52e11iT^{2} \)
37 \( 1 + (1.77e6 - 1.77e6i)T - 3.51e12iT^{2} \)
41 \( 1 + (-7.25e4 + 7.25e4i)T - 7.98e12iT^{2} \)
43 \( 1 + (-1.58e6 - 1.58e6i)T + 1.16e13iT^{2} \)
47 \( 1 + (-6.68e6 + 6.68e6i)T - 2.38e13iT^{2} \)
53 \( 1 + 1.04e7T + 6.22e13T^{2} \)
59 \( 1 - 3.32e6T + 1.46e14T^{2} \)
61 \( 1 + (6.53e6 + 6.53e6i)T + 1.91e14iT^{2} \)
67 \( 1 + 1.74e6iT - 4.06e14T^{2} \)
71 \( 1 + 4.04e6iT - 6.45e14T^{2} \)
73 \( 1 + (9.71e6 - 9.71e6i)T - 8.06e14iT^{2} \)
79 \( 1 + (3.20e7 + 3.20e7i)T + 1.51e15iT^{2} \)
83 \( 1 + 6.20e6T + 2.25e15T^{2} \)
89 \( 1 + (3.84e7 + 3.84e7i)T + 3.93e15iT^{2} \)
97 \( 1 + (2.17e7 - 2.17e7i)T - 7.83e15iT^{2} \)
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

−14.20467763858951515100413527870, −12.60000529320631503034435991913, −11.74455211176496806078952994471, −10.23046062030726643074255352618, −9.195116162018386953281850256660, −8.204982753398002228608599383590, −5.79181372514104189412905179060, −3.48237139024267017290872974401, −1.45337302459164127485980451076, −0.04132360941497004759952780250, 2.78037468348804526639817984790, 5.56386052467723632392534624099, 7.29865044743602599716493364168, 7.82510323085000609326639139881, 9.759929146381592527323196050247, 10.58085021765527666849186846525, 12.60403898328681951561923736086, 14.10905127351431512514755330970, 15.51023376485987398040236436900, 16.02566754021576193846512523512

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