↑

Properties

Label 2-912-19.11-c1-0-4
Degree $2$
Conductor $912$
Sign $-0.204 - 0.978i$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
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  + (−0.5 + 0.866i)3-s + (−1.53 + 2.65i)5-s + 2.06·7-s + (−0.499 − 0.866i)9-s + 6.45·11-s + (−0.5 − 0.866i)13-s + (−1.53 − 2.65i)15-s + (−0.694 + 1.20i)17-s + (3.75 + 2.20i)19-s + (−1.03 + 1.78i)21-s + (1.53 + 2.65i)23-s + (−2.19 − 3.80i)25-s + 0.999·27-s + (1.75 + 3.04i)29-s − 9.45·31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.685 + 1.18i)5-s + 0.780·7-s + (−0.166 − 0.288i)9-s + 1.94·11-s + (−0.138 − 0.240i)13-s + (−0.395 − 0.685i)15-s + (−0.168 + 0.291i)17-s + (0.862 + 0.506i)19-s + (−0.225 + 0.390i)21-s + (0.319 + 0.553i)23-s + (−0.438 − 0.760i)25-s + 0.192·27-s + (0.326 + 0.565i)29-s − 1.69·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.204 - 0.978i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.204 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.894560 + 1.10024i\)
\(L(\frac12)\) \(\approx\) \(0.894560 + 1.10024i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-3.75 - 2.20i)T \)
good5 \( 1 + (1.53 - 2.65i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 2.06T + 7T^{2} \)
11 \( 1 - 6.45T + 11T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.694 - 1.20i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.53 - 2.65i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.75 - 3.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.45T + 31T^{2} \)
37 \( 1 + 2.38T + 37T^{2} \)
41 \( 1 + (5.06 - 8.77i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.03 + 5.25i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.29 - 9.16i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.59 - 9.69i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.56 + 4.44i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.72 - 2.99i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.36 + 5.83i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.56 + 7.90i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.790 - 1.36i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 17.6T + 83T^{2} \)
89 \( 1 + (5.22 + 9.05i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.36 - 5.83i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.53521771796487140739002700105, −9.512913777954431685211686285525, −8.765274558076639114285327115933, −7.63385419895044787878405315464, −6.97507051995177819760461505104, −6.09629686738407887933307572120, −4.97817212847626108765272445169, −3.84967556953359069040535674931, −3.31605606642929933631129634443, −1.53726452050467517688175607532, 0.794530164684294562167648622508, 1.79808535618597171811345386664, 3.68291326029135179920936072981, 4.58938102270009655713289899411, 5.30188113725349336541977541325, 6.55369439710504472903558875664, 7.30711364660930618114147603265, 8.251406592853657326929923464438, 8.940940973971914431753629248336, 9.555616995774300742811869445548

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