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Properties

Label 2-804-201.5-c1-0-10
Degree $2$
Conductor $804$
Sign $0.998 - 0.0458i$
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  + (1.48 − 0.895i)3-s + (−2.22 + 0.652i)5-s + (2.01 + 1.74i)7-s + (1.39 − 2.65i)9-s + (−0.819 + 0.240i)11-s + (1.65 + 2.57i)13-s + (−2.71 + 2.95i)15-s + (5.09 + 0.732i)17-s + (0.289 + 0.333i)19-s + (4.55 + 0.785i)21-s + (7.43 − 3.39i)23-s + (0.311 − 0.200i)25-s + (−0.308 − 5.18i)27-s + 2.95i·29-s + (−0.121 + 0.189i)31-s + ⋯
L(s)  = 1  + (0.855 − 0.517i)3-s + (−0.994 + 0.291i)5-s + (0.762 + 0.660i)7-s + (0.465 − 0.885i)9-s + (−0.247 + 0.0725i)11-s + (0.458 + 0.713i)13-s + (−0.700 + 0.764i)15-s + (1.23 + 0.177i)17-s + (0.0663 + 0.0765i)19-s + (0.994 + 0.171i)21-s + (1.54 − 0.707i)23-s + (0.0623 − 0.0400i)25-s + (−0.0593 − 0.998i)27-s + 0.548i·29-s + (−0.0218 + 0.0340i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.998 - 0.0458i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ 0.998 - 0.0458i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98761 + 0.0455768i\)
\(L(\frac12)\) \(\approx\) \(1.98761 + 0.0455768i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 + 0.895i)T \)
67 \( 1 + (4.57 + 6.78i)T \)
good5 \( 1 + (2.22 - 0.652i)T + (4.20 - 2.70i)T^{2} \)
7 \( 1 + (-2.01 - 1.74i)T + (0.996 + 6.92i)T^{2} \)
11 \( 1 + (0.819 - 0.240i)T + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (-1.65 - 2.57i)T + (-5.40 + 11.8i)T^{2} \)
17 \( 1 + (-5.09 - 0.732i)T + (16.3 + 4.78i)T^{2} \)
19 \( 1 + (-0.289 - 0.333i)T + (-2.70 + 18.8i)T^{2} \)
23 \( 1 + (-7.43 + 3.39i)T + (15.0 - 17.3i)T^{2} \)
29 \( 1 - 2.95iT - 29T^{2} \)
31 \( 1 + (0.121 - 0.189i)T + (-12.8 - 28.1i)T^{2} \)
37 \( 1 - 1.16T + 37T^{2} \)
41 \( 1 + (0.236 - 1.64i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (-0.275 - 0.0395i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 + (-9.50 + 4.34i)T + (30.7 - 35.5i)T^{2} \)
53 \( 1 + (-0.673 - 4.68i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (5.71 - 8.88i)T + (-24.5 - 53.6i)T^{2} \)
61 \( 1 + (-0.773 + 2.63i)T + (-51.3 - 32.9i)T^{2} \)
71 \( 1 + (9.96 - 1.43i)T + (68.1 - 20.0i)T^{2} \)
73 \( 1 + (9.73 + 2.85i)T + (61.4 + 39.4i)T^{2} \)
79 \( 1 + (-4.58 - 7.13i)T + (-32.8 + 71.8i)T^{2} \)
83 \( 1 + (2.86 + 9.75i)T + (-69.8 + 44.8i)T^{2} \)
89 \( 1 + (1.51 + 0.690i)T + (58.2 + 67.2i)T^{2} \)
97 \( 1 + 10.4iT - 97T^{2} \)
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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

−10.29131801970295659400053378917, −9.023308537018630737585523787429, −8.599258774395032968774878288508, −7.65329669642982879351764928375, −7.18332533472208643254686736984, −5.97434196119264285033249991428, −4.72155050149770085021315038219, −3.65031848403212684500712101767, −2.72282765278794700031770059913, −1.36314764755361361200951331790, 1.14844414194674698876165039474, 2.94804321843432903277525903063, 3.80204565772681770812030987721, 4.65198072548276478972628545479, 5.55629073975466956893188607874, 7.34163250314261834323991458130, 7.74528716053240350825183656586, 8.415700649266681205006380235083, 9.319300453792524764957981527184, 10.31167507543789044259130604627

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