Properties

Label 2-684-19.18-c6-0-28
Degree $2$
Conductor $684$
Sign $-0.635 + 0.772i$
Analytic cond. $157.356$
Root an. cond. $12.5442$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 103.·5-s − 619.·7-s + 57.6·11-s + 481. i·13-s + 2.33e3·17-s + (4.35e3 − 5.29e3i)19-s + 8.69e3·23-s − 4.92e3·25-s + 4.60e4i·29-s + 3.67e4i·31-s + 6.40e4·35-s + 4.04e4i·37-s + 6.15e4i·41-s − 6.43e4·43-s + 9.39e4·47-s + ⋯
L(s)  = 1  − 0.827·5-s − 1.80·7-s + 0.0432·11-s + 0.219i·13-s + 0.476·17-s + (0.635 − 0.772i)19-s + 0.714·23-s − 0.315·25-s + 1.88i·29-s + 1.23i·31-s + 1.49·35-s + 0.798i·37-s + 0.892i·41-s − 0.809·43-s + 0.905·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 + 0.772i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.635 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.635 + 0.772i$
Analytic conductor: \(157.356\)
Root analytic conductor: \(12.5442\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :3),\ -0.635 + 0.772i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.1168000457\)
\(L(\frac12)\) \(\approx\) \(0.1168000457\)
\(L(4)\) 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 \( 1 \)
19 \( 1 + (-4.35e3 + 5.29e3i)T \)
good5 \( 1 + 103.T + 1.56e4T^{2} \)
7 \( 1 + 619.T + 1.17e5T^{2} \)
11 \( 1 - 57.6T + 1.77e6T^{2} \)
13 \( 1 - 481. iT - 4.82e6T^{2} \)
17 \( 1 - 2.33e3T + 2.41e7T^{2} \)
23 \( 1 - 8.69e3T + 1.48e8T^{2} \)
29 \( 1 - 4.60e4iT - 5.94e8T^{2} \)
31 \( 1 - 3.67e4iT - 8.87e8T^{2} \)
37 \( 1 - 4.04e4iT - 2.56e9T^{2} \)
41 \( 1 - 6.15e4iT - 4.75e9T^{2} \)
43 \( 1 + 6.43e4T + 6.32e9T^{2} \)
47 \( 1 - 9.39e4T + 1.07e10T^{2} \)
53 \( 1 - 9.78e4iT - 2.21e10T^{2} \)
59 \( 1 + 1.32e5iT - 4.21e10T^{2} \)
61 \( 1 + 1.92e3T + 5.15e10T^{2} \)
67 \( 1 - 3.78e5iT - 9.04e10T^{2} \)
71 \( 1 + 4.51e5iT - 1.28e11T^{2} \)
73 \( 1 + 6.69e5T + 1.51e11T^{2} \)
79 \( 1 - 7.47e5iT - 2.43e11T^{2} \)
83 \( 1 - 4.95e5T + 3.26e11T^{2} \)
89 \( 1 + 1.35e6iT - 4.96e11T^{2} \)
97 \( 1 + 7.59e4iT - 8.32e11T^{2} \)
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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

−9.256911121583586675567438660649, −8.487337433193803164389398989330, −7.18199150969952864920705057690, −6.85028909467356050975727970735, −5.70107720198165536167658794702, −4.59039745697829376699109497428, −3.33112158016913325017355726633, −3.03957143162234349480338431496, −1.18621931000747620099253272479, −0.03549545877075739874862473386, 0.71373002573891803641291108645, 2.45861985805907694495967273491, 3.50191829010418834532498337208, 4.03380138243028353419337080765, 5.57408931895329874739008556814, 6.28965681712601665845493827109, 7.32342066380330798915458710624, 7.955580098371709386358709021120, 9.169819876091509385407320337488, 9.791981112730930892586788708426

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