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Properties

Label 2-684-19.12-c4-0-30
Degree $2$
Conductor $684$
Sign $-0.983 - 0.180i$
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  + (−15.5 − 27.0i)5-s − 2.67·7-s − 42.3·11-s + (113. + 65.6i)13-s + (−143. − 247. i)17-s + (326. − 153. i)19-s + (158. − 273. i)23-s + (−173. + 300. i)25-s + (188. + 108. i)29-s − 475. i·31-s + (41.7 + 72.3i)35-s − 1.74e3i·37-s + (104. − 60.5i)41-s + (−1.23e3 − 2.14e3i)43-s + (−168. + 292. i)47-s + ⋯
L(s)  = 1  + (−0.623 − 1.08i)5-s − 0.0546·7-s − 0.349·11-s + (0.673 + 0.388i)13-s + (−0.495 − 0.858i)17-s + (0.904 − 0.426i)19-s + (0.299 − 0.517i)23-s + (−0.277 + 0.480i)25-s + (0.224 + 0.129i)29-s − 0.495i·31-s + (0.0340 + 0.0590i)35-s − 1.27i·37-s + (0.0623 − 0.0360i)41-s + (−0.669 − 1.15i)43-s + (−0.0763 + 0.132i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.180i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.983 - 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.983 - 0.180i$
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: \((2,\ 684,\ (\ :2),\ -0.983 - 0.180i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.6478210626\)
\(L(\frac12)\) \(\approx\) \(0.6478210626\)
\(L(3)\) 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 \( 1 + (-326. + 153. i)T \)
good5 \( 1 + (15.5 + 27.0i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + 2.67T + 2.40e3T^{2} \)
11 \( 1 + 42.3T + 1.46e4T^{2} \)
13 \( 1 + (-113. - 65.6i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (143. + 247. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (-158. + 273. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-188. - 108. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + 475. iT - 9.23e5T^{2} \)
37 \( 1 + 1.74e3iT - 1.87e6T^{2} \)
41 \( 1 + (-104. + 60.5i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (1.23e3 + 2.14e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (168. - 292. i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (-3.65e3 - 2.11e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (4.91e3 - 2.83e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (1.94e3 - 3.37e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-2.80e3 - 1.61e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (4.05e3 - 2.33e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (-803. - 1.39e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (2.90e3 - 1.68e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + 6.54e3T + 4.74e7T^{2} \)
89 \( 1 + (3.17e3 + 1.83e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (-4.87e3 + 2.81e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−9.128438968992560693300919034445, −8.766489956860795898194437943524, −7.73019452959734960531351076396, −6.92327084988982431756722072473, −5.66364308833924319524629387005, −4.78420004442315912113903029721, −3.97693847147939165622734729755, −2.67594897955325458371909902085, −1.16689082057564013649520690047, −0.17020551578443556507631982736, 1.47286391001940906786438568702, 3.00918012804843008033969072309, 3.57779985453723256761859131092, 4.87430290500203107420284450054, 6.06818624264205265743464327692, 6.82426489866846451992810075833, 7.78523663872593653457278363289, 8.402688552188347625094274736579, 9.645361276232223921131966499277, 10.46671981380297685396426202805

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