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Properties

Label 2-804-201.164-c1-0-8
Degree $2$
Conductor $804$
Sign $0.124 - 0.992i$
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  + (0.870 + 1.49i)3-s + 2.31·5-s + (−0.381 − 0.220i)7-s + (−1.48 + 2.60i)9-s + (−1.81 + 3.13i)11-s + (1.86 − 1.07i)13-s + (2.01 + 3.46i)15-s + (−3.58 + 2.07i)17-s + (2.79 + 4.83i)19-s + (−0.00221 − 0.763i)21-s + (3.74 − 2.16i)23-s + 0.350·25-s + (−5.19 + 0.0453i)27-s + (−1.35 − 0.783i)29-s + (9.56 + 5.52i)31-s + ⋯
L(s)  = 1  + (0.502 + 0.864i)3-s + 1.03·5-s + (−0.144 − 0.0832i)7-s + (−0.494 + 0.868i)9-s + (−0.546 + 0.946i)11-s + (0.516 − 0.298i)13-s + (0.519 + 0.894i)15-s + (−0.869 + 0.502i)17-s + (0.641 + 1.11i)19-s + (−0.000484 − 0.166i)21-s + (0.780 − 0.450i)23-s + 0.0700·25-s + (−0.999 + 0.00872i)27-s + (−0.251 − 0.145i)29-s + (1.71 + 0.991i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.124 - 0.992i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (365, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ 0.124 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51949 + 1.34119i\)
\(L(\frac12)\) \(\approx\) \(1.51949 + 1.34119i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.870 - 1.49i)T \)
67 \( 1 + (-7.39 - 3.51i)T \)
good5 \( 1 - 2.31T + 5T^{2} \)
7 \( 1 + (0.381 + 0.220i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.81 - 3.13i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.86 + 1.07i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.58 - 2.07i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.79 - 4.83i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.74 + 2.16i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.35 + 0.783i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-9.56 - 5.52i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.08 - 3.60i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.96 + 6.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 10.1iT - 43T^{2} \)
47 \( 1 + (1.58 + 0.914i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 5.10T + 53T^{2} \)
59 \( 1 - 11.2iT - 59T^{2} \)
61 \( 1 + (-4.53 + 2.61i)T + (30.5 - 52.8i)T^{2} \)
71 \( 1 + (8.03 + 4.64i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.45 + 5.98i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-9.57 - 5.52i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (15.2 - 8.82i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 1.87iT - 89T^{2} \)
97 \( 1 + (-3.26 + 1.88i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.19542834508290977771277947832, −9.833965478891133609108320611599, −8.843905375812847747248141026601, −8.149723569262581218211547280956, −6.98549772794669284132208310033, −5.89463132495256168904533558363, −5.08576732026565910983926334481, −4.10110638991269569259750958909, −2.90557927440775316228473131426, −1.86001644744451834019953771636, 0.996913499768895295259734975674, 2.41112334937249143578997518364, 3.12590709731553020139695537989, 4.76719184604646933068073071143, 5.98141000206860798511617007957, 6.43473098677684051068907743019, 7.51111894247402158520158462883, 8.390929340093556524818513854583, 9.263646239570702119919770879755, 9.718311410531430310250835989957

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