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Properties

Label 2-684-19.12-c4-0-11
Degree $2$
Conductor $684$
Sign $0.407 - 0.913i$
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  + (3.11 + 5.39i)5-s + 49.9·7-s − 1.88·11-s + (−69.0 − 39.8i)13-s + (119. + 207. i)17-s + (72.4 + 353. i)19-s + (−109. + 189. i)23-s + (293. − 507. i)25-s + (−340. − 196. i)29-s + 580. i·31-s + (155. + 269. i)35-s + 2.47e3i·37-s + (2.84e3 − 1.64e3i)41-s + (−162. − 282. i)43-s + (969. − 1.67e3i)47-s + ⋯
L(s)  = 1  + (0.124 + 0.215i)5-s + 1.01·7-s − 0.0155·11-s + (−0.408 − 0.235i)13-s + (0.414 + 0.717i)17-s + (0.200 + 0.979i)19-s + (−0.206 + 0.358i)23-s + (0.468 − 0.812i)25-s + (−0.404 − 0.233i)29-s + 0.604i·31-s + (0.126 + 0.219i)35-s + 1.81i·37-s + (1.69 − 0.975i)41-s + (−0.0881 − 0.152i)43-s + (0.438 − 0.760i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.407 - 0.913i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.407 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.407 - 0.913i$
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: \((2,\ 684,\ (\ :2),\ 0.407 - 0.913i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.248221892\)
\(L(\frac12)\) \(\approx\) \(2.248221892\)
\(L(3)\) 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 \( 1 + (-72.4 - 353. i)T \)
good5 \( 1 + (-3.11 - 5.39i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 - 49.9T + 2.40e3T^{2} \)
11 \( 1 + 1.88T + 1.46e4T^{2} \)
13 \( 1 + (69.0 + 39.8i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (-119. - 207. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (109. - 189. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (340. + 196. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 - 580. iT - 9.23e5T^{2} \)
37 \( 1 - 2.47e3iT - 1.87e6T^{2} \)
41 \( 1 + (-2.84e3 + 1.64e3i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (162. + 282. i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-969. + 1.67e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (2.08e3 + 1.20e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (3.04e3 - 1.75e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (1.03e3 - 1.79e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (4.38e3 + 2.53e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (2.99e3 - 1.72e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (-4.35e3 - 7.54e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-3.64e3 + 2.10e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 1.17e4T + 4.74e7T^{2} \)
89 \( 1 + (-8.21e3 - 4.74e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (-5.04e3 + 2.91e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−10.25147460284775129805074734511, −9.178531903733562001398766753176, −8.130327644856153536469982604589, −7.67203583059482731235533296537, −6.45217785197568680650614974571, −5.52757548006592042892706592817, −4.60848661953528693325143631814, −3.49872199933246113988146225392, −2.19842771049765408633441933867, −1.13474158677476502675431485162, 0.58464106164446580432702072551, 1.80992290560620567854379756715, 2.95331874234824543854486396936, 4.41158123650731781149572645135, 5.05693592601547435730741079467, 6.07098576507130275986443732227, 7.35534271987276783008283129189, 7.81645667820754948058109628593, 9.094023807100316389349655319086, 9.462790745145748646447892466962

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