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Properties

Label 2-804-12.11-c1-0-104
Degree $2$
Conductor $804$
Sign $-0.624 + 0.781i$
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  + (−0.743 + 1.20i)2-s + (−1.57 + 0.727i)3-s + (−0.893 − 1.78i)4-s − 4.19i·5-s + (0.293 − 2.43i)6-s − 2.21i·7-s + (2.81 + 0.255i)8-s + (1.94 − 2.28i)9-s + (5.04 + 3.12i)10-s − 2.03·11-s + (2.70 + 2.16i)12-s + 2.41·13-s + (2.66 + 1.64i)14-s + (3.05 + 6.59i)15-s + (−2.40 + 3.19i)16-s − 3.05i·17-s + ⋯
L(s)  = 1  + (−0.525 + 0.850i)2-s + (−0.907 + 0.420i)3-s + (−0.446 − 0.894i)4-s − 1.87i·5-s + (0.119 − 0.992i)6-s − 0.837i·7-s + (0.995 + 0.0903i)8-s + (0.646 − 0.762i)9-s + (1.59 + 0.987i)10-s − 0.612·11-s + (0.781 + 0.624i)12-s + 0.669·13-s + (0.712 + 0.440i)14-s + (0.788 + 1.70i)15-s + (−0.600 + 0.799i)16-s − 0.742i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.624 + 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $-0.624 + 0.781i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (671, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ -0.624 + 0.781i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.199403 - 0.414412i\)
\(L(\frac12)\) \(\approx\) \(0.199403 - 0.414412i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.743 - 1.20i)T \)
3 \( 1 + (1.57 - 0.727i)T \)
67 \( 1 + iT \)
good5 \( 1 + 4.19iT - 5T^{2} \)
7 \( 1 + 2.21iT - 7T^{2} \)
11 \( 1 + 2.03T + 11T^{2} \)
13 \( 1 - 2.41T + 13T^{2} \)
17 \( 1 + 3.05iT - 17T^{2} \)
19 \( 1 - 1.61iT - 19T^{2} \)
23 \( 1 + 4.01T + 23T^{2} \)
29 \( 1 + 4.93iT - 29T^{2} \)
31 \( 1 + 8.17iT - 31T^{2} \)
37 \( 1 - 4.89T + 37T^{2} \)
41 \( 1 - 10.7iT - 41T^{2} \)
43 \( 1 - 5.11iT - 43T^{2} \)
47 \( 1 - 4.92T + 47T^{2} \)
53 \( 1 - 6.70iT - 53T^{2} \)
59 \( 1 + 10.1T + 59T^{2} \)
61 \( 1 + 13.2T + 61T^{2} \)
71 \( 1 - 5.85T + 71T^{2} \)
73 \( 1 + 7.24T + 73T^{2} \)
79 \( 1 + 13.0iT - 79T^{2} \)
83 \( 1 - 5.22T + 83T^{2} \)
89 \( 1 - 7.67iT - 89T^{2} \)
97 \( 1 + 10.4T + 97T^{2} \)
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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

−9.647268076193778802921831704025, −9.315436568297599040514632514998, −8.075517372947920338678477622615, −7.64688893283637523398279715573, −6.16209047846787001946935344878, −5.68195571430633141318566889082, −4.54466850388377883621773257776, −4.28197985126921847140813809880, −1.30606876575478798830859855950, −0.33710773354127399246767967848, 1.87389766378383481039904276530, 2.79463627911004793306137222115, 3.86506667967143614741731355096, 5.39606666605960775039853482504, 6.33126650349250705070063547762, 7.14411665583815586124866593053, 7.935403035859444682612496544385, 8.975356278280148610670496010043, 10.26260056167843913766430582227, 10.61051912996247973047342015954

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