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Properties

Label 2-684-19.12-c4-0-14
Degree $2$
Conductor $684$
Sign $0.103 + 0.994i$
Analytic cond. $70.7050$
Root an. cond. $8.40862$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−21.2 − 36.7i)5-s − 85.9·7-s + 212.·11-s + (112. + 65.1i)13-s + (17.9 + 31.0i)17-s + (239. + 270. i)19-s + (78.2 − 135. i)23-s + (−586. + 1.01e3i)25-s + (1.13e3 + 656. i)29-s + 722. i·31-s + (1.82e3 + 3.15e3i)35-s − 2.08e3i·37-s + (−1.38e3 + 800. i)41-s + (−493. − 855. i)43-s + (817. − 1.41e3i)47-s + ⋯
L(s)  = 1  + (−0.848 − 1.46i)5-s − 1.75·7-s + 1.75·11-s + (0.667 + 0.385i)13-s + (0.0619 + 0.107i)17-s + (0.663 + 0.748i)19-s + (0.147 − 0.256i)23-s + (−0.938 + 1.62i)25-s + (1.35 + 0.780i)29-s + 0.751i·31-s + (1.48 + 2.57i)35-s − 1.52i·37-s + (−0.824 + 0.476i)41-s + (−0.266 − 0.462i)43-s + (0.370 − 0.640i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.103 + 0.994i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.103 + 0.994i$
Analytic conductor: \(70.7050\)
Root analytic conductor: \(8.40862\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :2),\ 0.103 + 0.994i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.408508580\)
\(L(\frac12)\) \(\approx\) \(1.408508580\)
\(L(3)\) 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 \( 1 + (-239. - 270. i)T \)
good5 \( 1 + (21.2 + 36.7i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + 85.9T + 2.40e3T^{2} \)
11 \( 1 - 212.T + 1.46e4T^{2} \)
13 \( 1 + (-112. - 65.1i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (-17.9 - 31.0i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (-78.2 + 135. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-1.13e3 - 656. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 - 722. iT - 9.23e5T^{2} \)
37 \( 1 + 2.08e3iT - 1.87e6T^{2} \)
41 \( 1 + (1.38e3 - 800. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (493. + 855. i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-817. + 1.41e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (2.58e3 + 1.49e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-486. + 280. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-1.18e3 + 2.04e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (5.30e3 + 3.06e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (-7.17e3 + 4.14e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (-2.25e3 - 3.90e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-1.05e4 + 6.11e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + 3.73e3T + 4.74e7T^{2} \)
89 \( 1 + (1.17e3 + 676. i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (-1.84e3 + 1.06e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−9.277535447437152628025786384025, −9.092712516926460967388653957291, −8.155862761470631379444547871454, −6.86779020497914604732071217610, −6.28263056521834339189801377366, −5.05496476245289799096140374865, −3.85422778463078573191812629303, −3.49168151683913688421153784965, −1.41214726247927807319857837310, −0.50630815269577072398214358383, 0.825497942243964577408058686226, 2.85688934158590082282526848607, 3.37090166007385108576588863151, 4.19916065158474631229736948127, 6.09453082024438247143684598122, 6.61021968212873889641044665700, 7.14827564479536009628502079371, 8.344556321767376596845310339497, 9.469179137642330695890055738898, 9.988806358306958633338446327194

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