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Properties

Label 2-684-19.12-c4-0-15
Degree $2$
Conductor $684$
Sign $0.976 - 0.216i$
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  + (−8.78 − 15.2i)5-s + 69.3·7-s + 198.·11-s + (215. + 124. i)13-s + (112. + 194. i)17-s + (240. − 268. i)19-s + (−462. + 800. i)23-s + (158. − 274. i)25-s + (−964. − 557. i)29-s + 1.35e3i·31-s + (−609. − 1.05e3i)35-s + 917. i·37-s + (620. − 358. i)41-s + (−806. − 1.39e3i)43-s + (−1.96e3 + 3.40e3i)47-s + ⋯
L(s)  = 1  + (−0.351 − 0.608i)5-s + 1.41·7-s + 1.64·11-s + (1.27 + 0.734i)13-s + (0.388 + 0.672i)17-s + (0.667 − 0.744i)19-s + (−0.874 + 1.51i)23-s + (0.253 − 0.438i)25-s + (−1.14 − 0.662i)29-s + 1.41i·31-s + (−0.497 − 0.861i)35-s + 0.670i·37-s + (0.369 − 0.213i)41-s + (−0.436 − 0.755i)43-s + (−0.890 + 1.54i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.216i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.976 - 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.976 - 0.216i$
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: \((2,\ 684,\ (\ :2),\ 0.976 - 0.216i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.992943720\)
\(L(\frac12)\) \(\approx\) \(2.992943720\)
\(L(3)\) 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 \( 1 + (-240. + 268. i)T \)
good5 \( 1 + (8.78 + 15.2i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 - 69.3T + 2.40e3T^{2} \)
11 \( 1 - 198.T + 1.46e4T^{2} \)
13 \( 1 + (-215. - 124. i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (-112. - 194. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (462. - 800. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (964. + 557. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 - 1.35e3iT - 9.23e5T^{2} \)
37 \( 1 - 917. iT - 1.87e6T^{2} \)
41 \( 1 + (-620. + 358. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (806. + 1.39e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (1.96e3 - 3.40e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (-1.55e3 - 900. i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-3.31e3 + 1.91e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-239. + 414. i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-3.19e3 - 1.84e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (4.28e3 - 2.47e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (-1.55e3 - 2.69e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (4.95e3 - 2.86e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 6.05e3T + 4.74e7T^{2} \)
89 \( 1 + (1.19e4 + 6.92e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (-1.50e4 + 8.70e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−9.773842832789517984477740058964, −8.857457298997961869328540477999, −8.376269086052222938747154124427, −7.40739367264067267171348991375, −6.34391521213709106853936838968, −5.33149528917551531073805870264, −4.28042793166342340692365202189, −3.66262803976032837125429577580, −1.62487756119350418352296525263, −1.19614526501071011383613240564, 0.869033202173439021057987599750, 1.87301052761630505352891972890, 3.43851586489039946662827141437, 4.14330137284338326941265381805, 5.39412716494450726081403216021, 6.30634550156937776933119877459, 7.34091288189299352776939884465, 8.114036350662267837628120043429, 8.857298843531229382784526058045, 9.926342806590802040932855573104

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