Properties

Label 2-804-67.54-c1-0-4
Degree $2$
Conductor $804$
Sign $0.955 + 0.295i$
Analytic cond. $6.41997$
Root an. cond. $2.53376$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.654 − 0.755i)3-s + (−2.40 + 1.54i)5-s + (−2.21 − 0.887i)7-s + (−0.142 − 0.989i)9-s + (0.0929 + 1.95i)11-s + (6.57 − 0.627i)13-s + (−0.407 + 2.83i)15-s + (−0.498 − 2.05i)17-s + (6.01 − 2.40i)19-s + (−2.12 + 1.09i)21-s + (5.97 + 1.15i)23-s + (1.32 − 2.90i)25-s + (−0.841 − 0.540i)27-s + (3.14 − 5.44i)29-s + (6.99 + 0.668i)31-s + ⋯
L(s)  = 1  + (0.378 − 0.436i)3-s + (−1.07 + 0.692i)5-s + (−0.837 − 0.335i)7-s + (−0.0474 − 0.329i)9-s + (0.0280 + 0.588i)11-s + (1.82 − 0.174i)13-s + (−0.105 + 0.731i)15-s + (−0.120 − 0.498i)17-s + (1.37 − 0.552i)19-s + (−0.462 + 0.238i)21-s + (1.24 + 0.239i)23-s + (0.265 − 0.581i)25-s + (−0.161 − 0.104i)27-s + (0.583 − 1.01i)29-s + (1.25 + 0.119i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.955 + 0.295i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ 0.955 + 0.295i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42120 - 0.215022i\)
\(L(\frac12)\) \(\approx\) \(1.42120 - 0.215022i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.654 + 0.755i)T \)
67 \( 1 + (-2.37 + 7.83i)T \)
good5 \( 1 + (2.40 - 1.54i)T + (2.07 - 4.54i)T^{2} \)
7 \( 1 + (2.21 + 0.887i)T + (5.06 + 4.83i)T^{2} \)
11 \( 1 + (-0.0929 - 1.95i)T + (-10.9 + 1.04i)T^{2} \)
13 \( 1 + (-6.57 + 0.627i)T + (12.7 - 2.46i)T^{2} \)
17 \( 1 + (0.498 + 2.05i)T + (-15.1 + 7.78i)T^{2} \)
19 \( 1 + (-6.01 + 2.40i)T + (13.7 - 13.1i)T^{2} \)
23 \( 1 + (-5.97 - 1.15i)T + (21.3 + 8.54i)T^{2} \)
29 \( 1 + (-3.14 + 5.44i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.99 - 0.668i)T + (30.4 + 5.86i)T^{2} \)
37 \( 1 + (-0.506 - 0.877i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.25 - 2.14i)T + (1.95 - 40.9i)T^{2} \)
43 \( 1 + (4.07 + 1.19i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 + (-2.70 - 7.82i)T + (-36.9 + 29.0i)T^{2} \)
53 \( 1 + (5.19 - 1.52i)T + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (0.533 + 1.16i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (0.212 - 4.45i)T + (-60.7 - 5.79i)T^{2} \)
71 \( 1 + (1.57 - 6.51i)T + (-63.1 - 32.5i)T^{2} \)
73 \( 1 + (0.0356 - 0.748i)T + (-72.6 - 6.93i)T^{2} \)
79 \( 1 + (3.92 + 5.51i)T + (-25.8 + 74.6i)T^{2} \)
83 \( 1 + (-9.09 - 4.68i)T + (48.1 + 67.6i)T^{2} \)
89 \( 1 + (9.03 + 10.4i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (5.11 + 8.85i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.20204109744326085311655492668, −9.348038718498379467297119309849, −8.394717399054206255278809271821, −7.54161237018569326520557276652, −6.89947012394651747574131865942, −6.15562082335854101547712000793, −4.61663446657243729272109379462, −3.44174671342398725414766774478, −2.97958892116633965256085797318, −0.965347509662727587899732814237, 1.06509766667199405971251628009, 3.20541691590064249677091908463, 3.64552977585511994234855673842, 4.78829315286601952495098914045, 5.87027093436103615724636334099, 6.82791067325373969844866689669, 8.106685982896289793726440798768, 8.581300883952541485348234463659, 9.227476373350521943278844113027, 10.29417537410824398891598873100

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