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Properties

Label 2-804-201.5-c1-0-16
Degree $2$
Conductor $804$
Sign $0.983 + 0.183i$
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  + (1.64 − 0.533i)3-s + (2.22 − 0.652i)5-s + (2.01 + 1.74i)7-s + (2.42 − 1.75i)9-s + (0.819 − 0.240i)11-s + (1.65 + 2.57i)13-s + (3.31 − 2.26i)15-s + (−5.09 − 0.732i)17-s + (0.289 + 0.333i)19-s + (4.25 + 1.80i)21-s + (−7.43 + 3.39i)23-s + (0.311 − 0.200i)25-s + (3.06 − 4.19i)27-s − 2.95i·29-s + (−0.121 + 0.189i)31-s + ⋯
L(s)  = 1  + (0.951 − 0.308i)3-s + (0.994 − 0.291i)5-s + (0.762 + 0.660i)7-s + (0.809 − 0.586i)9-s + (0.247 − 0.0725i)11-s + (0.458 + 0.713i)13-s + (0.855 − 0.584i)15-s + (−1.23 − 0.177i)17-s + (0.0663 + 0.0765i)19-s + (0.929 + 0.393i)21-s + (−1.54 + 0.707i)23-s + (0.0623 − 0.0400i)25-s + (0.589 − 0.807i)27-s − 0.548i·29-s + (−0.0218 + 0.0340i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.983 + 0.183i$
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: \((2,\ 804,\ (\ :1/2),\ 0.983 + 0.183i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.68392 - 0.247804i\)
\(L(\frac12)\) \(\approx\) \(2.68392 - 0.247804i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.64 + 0.533i)T \)
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

−9.822569410298668137118026313439, −9.375145445077280291977180234562, −8.559714627740280075634415441937, −7.938949990373700130231533250995, −6.69871015601179406031604146551, −5.96099349284164921668554766259, −4.80387568003857769248375180081, −3.74325699232976632736940370099, −2.20119806137987462427330830248, −1.72480687077710051995988154615, 1.61715343116727285643148270206, 2.56677755933730144969774130225, 3.88335335030113256079536521669, 4.69739268662777893875227066522, 5.93610579982990133894499537524, 6.88454802398943000195742219458, 7.914250784545629826098622344458, 8.566096197335222303196718728847, 9.462099895530842656858847828434, 10.36052051948295223685941871227

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