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Properties

Label 2-684-19.12-c4-0-29
Degree $2$
Conductor $684$
Sign $-0.997 + 0.0696i$
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  + (−12.7 − 22.1i)5-s − 23.8·7-s + 198.·11-s + (−233. − 134. i)13-s + (59.9 + 103. i)17-s + (278. − 230. i)19-s + (−25.4 + 44.1i)23-s + (−14.9 + 25.8i)25-s + (318. + 184. i)29-s − 1.89e3i·31-s + (304. + 528. i)35-s + 1.05e3i·37-s + (171. − 98.8i)41-s + (259. + 449. i)43-s + (422. − 732. i)47-s + ⋯
L(s)  = 1  + (−0.511 − 0.886i)5-s − 0.486·7-s + 1.63·11-s + (−1.38 − 0.797i)13-s + (0.207 + 0.359i)17-s + (0.770 − 0.637i)19-s + (−0.0481 + 0.0833i)23-s + (−0.0238 + 0.0413i)25-s + (0.378 + 0.218i)29-s − 1.96i·31-s + (0.248 + 0.431i)35-s + 0.771i·37-s + (0.101 − 0.0587i)41-s + (0.140 + 0.243i)43-s + (0.191 − 0.331i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0696i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.997 + 0.0696i$
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: \((2,\ 684,\ (\ :2),\ -0.997 + 0.0696i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.8149676583\)
\(L(\frac12)\) \(\approx\) \(0.8149676583\)
\(L(3)\) 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 \( 1 + (-278. + 230. i)T \)
good5 \( 1 + (12.7 + 22.1i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + 23.8T + 2.40e3T^{2} \)
11 \( 1 - 198.T + 1.46e4T^{2} \)
13 \( 1 + (233. + 134. i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (-59.9 - 103. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (25.4 - 44.1i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-318. - 184. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + 1.89e3iT - 9.23e5T^{2} \)
37 \( 1 - 1.05e3iT - 1.87e6T^{2} \)
41 \( 1 + (-171. + 98.8i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-259. - 449. i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-422. + 732. i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (382. + 220. i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-5.34e3 + 3.08e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (2.01e3 - 3.49e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (5.12e3 + 2.95e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (7.73e3 - 4.46e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (-47.6 - 82.4i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (5.17e3 - 2.98e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + 7.20e3T + 4.74e7T^{2} \)
89 \( 1 + (261. + 150. i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (1.02e4 - 5.91e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−9.506210737997199904190483070424, −8.670415911430021813229168443231, −7.74798170077571837851970354401, −6.89133580822078213054013731894, −5.83642578626183394124494573373, −4.77217417159551689401943447944, −3.96098905739480054419626470815, −2.78529218664737527583018767488, −1.20799635534760154544950674117, −0.21378301402308125872539091234, 1.41829635554435314980705113383, 2.85860652189763302583015159578, 3.69983842000406950699541846822, 4.71926651369464463279022202755, 6.05715330181432998579523137695, 7.05110151220075405915624727883, 7.28252624754883976967275153595, 8.732286098270490422854540348069, 9.523373613300765140055403104227, 10.20405963409588911844760795893

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