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Properties

Label 2-804-12.11-c1-0-108
Degree $2$
Conductor $804$
Sign $-0.481 + 0.876i$
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  + (−1.41 − 0.108i)2-s + (1.62 + 0.591i)3-s + (1.97 + 0.307i)4-s − 3.04i·5-s + (−2.23 − 1.01i)6-s + 0.278i·7-s + (−2.75 − 0.648i)8-s + (2.30 + 1.92i)9-s + (−0.332 + 4.29i)10-s − 5.07·11-s + (3.03 + 1.66i)12-s − 5.86·13-s + (0.0303 − 0.392i)14-s + (1.80 − 4.96i)15-s + (3.81 + 1.21i)16-s − 7.60i·17-s + ⋯
L(s)  = 1  + (−0.997 − 0.0769i)2-s + (0.939 + 0.341i)3-s + (0.988 + 0.153i)4-s − 1.36i·5-s + (−0.910 − 0.412i)6-s + 0.105i·7-s + (−0.973 − 0.229i)8-s + (0.766 + 0.642i)9-s + (−0.104 + 1.35i)10-s − 1.53·11-s + (0.876 + 0.481i)12-s − 1.62·13-s + (0.00810 − 0.105i)14-s + (0.465 − 1.28i)15-s + (0.952 + 0.303i)16-s − 1.84i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $-0.481 + 0.876i$
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: \((2,\ 804,\ (\ :1/2),\ -0.481 + 0.876i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.433642 - 0.733281i\)
\(L(\frac12)\) \(\approx\) \(0.433642 - 0.733281i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.108i)T \)
3 \( 1 + (-1.62 - 0.591i)T \)
67 \( 1 + iT \)
good5 \( 1 + 3.04iT - 5T^{2} \)
7 \( 1 - 0.278iT - 7T^{2} \)
11 \( 1 + 5.07T + 11T^{2} \)
13 \( 1 + 5.86T + 13T^{2} \)
17 \( 1 + 7.60iT - 17T^{2} \)
19 \( 1 + 5.91iT - 19T^{2} \)
23 \( 1 - 2.24T + 23T^{2} \)
29 \( 1 + 5.83iT - 29T^{2} \)
31 \( 1 - 4.34iT - 31T^{2} \)
37 \( 1 + 6.86T + 37T^{2} \)
41 \( 1 - 8.48iT - 41T^{2} \)
43 \( 1 + 2.79iT - 43T^{2} \)
47 \( 1 - 0.455T + 47T^{2} \)
53 \( 1 + 5.43iT - 53T^{2} \)
59 \( 1 + 0.908T + 59T^{2} \)
61 \( 1 - 1.16T + 61T^{2} \)
71 \( 1 + 4.54T + 71T^{2} \)
73 \( 1 - 13.5T + 73T^{2} \)
79 \( 1 + 12.3iT - 79T^{2} \)
83 \( 1 - 2.79T + 83T^{2} \)
89 \( 1 - 1.69iT - 89T^{2} \)
97 \( 1 - 4.26T + 97T^{2} \)
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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

−9.681594689441691658592965009495, −9.178217007218805350931200849568, −8.464311059481293377960451635870, −7.61530427625575699319355771958, −7.09241886176358971141475143188, −5.06911274030057366232156023010, −4.89406365913112426422981113309, −2.92656567936418041294390239467, −2.28741472195252301784908871421, −0.47289260170585713040687862656, 1.96262383294635790791577746928, 2.69400808046840278846665062555, 3.65144490005244122004037287390, 5.54270299209209978869308475263, 6.63840014409056111156982757752, 7.42560135211859870321286704008, 7.82934561654233578894306863206, 8.706316275583176514722083622575, 9.878213722296632599260611584832, 10.38317938999608179058408218663

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