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Properties

Label 2-684-19.18-c6-0-47
Degree $2$
Conductor $684$
Sign $0.147 + 0.989i$
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  + 186.·5-s + 202.·7-s + 1.59e3·11-s + 143. i·13-s − 4.77e3·17-s + (−1.01e3 − 6.78e3i)19-s − 7.73e3·23-s + 1.92e4·25-s − 8.82e3i·29-s − 5.15e4i·31-s + 3.77e4·35-s − 9.46e4i·37-s − 7.87e4i·41-s − 1.29e5·43-s + 9.72e4·47-s + ⋯
L(s)  = 1  + 1.49·5-s + 0.589·7-s + 1.19·11-s + 0.0652i·13-s − 0.971·17-s + (−0.147 − 0.989i)19-s − 0.635·23-s + 1.23·25-s − 0.361i·29-s − 1.73i·31-s + 0.881·35-s − 1.86i·37-s − 1.14i·41-s − 1.62·43-s + 0.936·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.147 + 0.989i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.147 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.147 + 0.989i$
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: \((2,\ 684,\ (\ :3),\ 0.147 + 0.989i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(3.061638988\)
\(L(\frac12)\) \(\approx\) \(3.061638988\)
\(L(4)\) 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 \( 1 + (1.01e3 + 6.78e3i)T \)
good5 \( 1 - 186.T + 1.56e4T^{2} \)
7 \( 1 - 202.T + 1.17e5T^{2} \)
11 \( 1 - 1.59e3T + 1.77e6T^{2} \)
13 \( 1 - 143. iT - 4.82e6T^{2} \)
17 \( 1 + 4.77e3T + 2.41e7T^{2} \)
23 \( 1 + 7.73e3T + 1.48e8T^{2} \)
29 \( 1 + 8.82e3iT - 5.94e8T^{2} \)
31 \( 1 + 5.15e4iT - 8.87e8T^{2} \)
37 \( 1 + 9.46e4iT - 2.56e9T^{2} \)
41 \( 1 + 7.87e4iT - 4.75e9T^{2} \)
43 \( 1 + 1.29e5T + 6.32e9T^{2} \)
47 \( 1 - 9.72e4T + 1.07e10T^{2} \)
53 \( 1 - 4.61e4iT - 2.21e10T^{2} \)
59 \( 1 + 6.17e4iT - 4.21e10T^{2} \)
61 \( 1 + 8.56e4T + 5.15e10T^{2} \)
67 \( 1 + 4.90e4iT - 9.04e10T^{2} \)
71 \( 1 + 4.08e5iT - 1.28e11T^{2} \)
73 \( 1 + 1.30e5T + 1.51e11T^{2} \)
79 \( 1 + 1.06e5iT - 2.43e11T^{2} \)
83 \( 1 - 1.22e4T + 3.26e11T^{2} \)
89 \( 1 - 5.61e5iT - 4.96e11T^{2} \)
97 \( 1 + 8.51e5iT - 8.32e11T^{2} \)
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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

−9.233524980453403337850366327824, −8.841159048574291742551257953858, −7.51632819929633662850250651845, −6.49409870767990108550874243657, −5.91194042259744426521753341389, −4.85408669321332528782324788233, −3.90406037576956561318611696613, −2.29622778340841991990040084914, −1.81929727253874461797916092725, −0.51535617332529586930681644926, 1.38522088712797030679703027530, 1.75901650389489246034224927829, 3.09215010281555230121685146001, 4.39401066919660289558235303062, 5.29424621851049872812936038394, 6.30293330209937689109220649784, 6.78014119189742161410905446828, 8.241220917361034010528357802807, 8.892093234239756219545535992863, 9.822262079480074038512306302394

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