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Properties

Label 2-2808-117.94-c1-0-8
Degree $2$
Conductor $2808$
Sign $0.0260 - 0.999i$
Analytic cond. $22.4219$
Root an. cond. $4.73518$
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  + (−0.766 + 1.32i)5-s − 3.51·7-s + (3.19 − 5.53i)11-s + (−3.46 − 1.00i)13-s + (0.550 − 0.953i)17-s + (−1.13 + 1.97i)19-s + 3.32·23-s + (1.32 + 2.29i)25-s + (−0.714 + 1.23i)29-s + (−1.93 + 3.35i)31-s + (2.69 − 4.66i)35-s + (1.81 + 3.14i)37-s − 11.9·41-s + 12.5·43-s + (1.23 + 2.13i)47-s + ⋯
L(s)  = 1  + (−0.342 + 0.594i)5-s − 1.32·7-s + (0.963 − 1.66i)11-s + (−0.960 − 0.279i)13-s + (0.133 − 0.231i)17-s + (−0.261 + 0.452i)19-s + 0.693·23-s + (0.264 + 0.458i)25-s + (−0.132 + 0.229i)29-s + (−0.347 + 0.602i)31-s + (0.455 − 0.789i)35-s + (0.298 + 0.517i)37-s − 1.87·41-s + 1.91·43-s + (0.179 + 0.311i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0260 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0260 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2808\)    =    \(2^{3} \cdot 3^{3} \cdot 13\)
Sign: $0.0260 - 0.999i$
Analytic conductor: \(22.4219\)
Root analytic conductor: \(4.73518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2808} (1225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2808,\ (\ :1/2),\ 0.0260 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8904554977\)
\(L(\frac12)\) \(\approx\) \(0.8904554977\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 + (3.46 + 1.00i)T \)
good5 \( 1 + (0.766 - 1.32i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 3.51T + 7T^{2} \)
11 \( 1 + (-3.19 + 5.53i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.550 + 0.953i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.13 - 1.97i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.32T + 23T^{2} \)
29 \( 1 + (0.714 - 1.23i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.93 - 3.35i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.81 - 3.14i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 11.9T + 41T^{2} \)
43 \( 1 - 12.5T + 43T^{2} \)
47 \( 1 + (-1.23 - 2.13i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6.18T + 53T^{2} \)
59 \( 1 + (-2.42 - 4.20i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 3.09T + 61T^{2} \)
67 \( 1 + 7.60T + 67T^{2} \)
71 \( 1 + (2.43 - 4.21i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 5.40T + 73T^{2} \)
79 \( 1 + (-2.39 - 4.14i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.19 - 5.53i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.25 - 7.36i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.52T + 97T^{2} \)
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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

−9.047774551012541705391086278829, −8.311991422368531676565107083311, −7.24622408848845352272028618326, −6.77597810426513340390720507982, −6.02838842219633091128188697773, −5.27931210686188670768272596542, −3.94637374557727225595324166079, −3.29480152264743273396547911266, −2.73193657042735935666224273772, −0.979572411654696205006073262328, 0.34470687528149499903669065246, 1.85096082562119844361164077914, 2.85613824268766643238851242971, 4.04429104224418652891939894473, 4.49088876549398521580099626109, 5.47508784810799574207115869685, 6.57940299625703731082702662220, 6.99407958332394339337959478055, 7.70906819205430849139975619165, 8.900317097357933690316413832531

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