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Properties

Label 2-684-19.12-c4-0-21
Degree $2$
Conductor $684$
Sign $0.136 + 0.990i$
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  + (−13.2 − 22.8i)5-s + 3.95·7-s + 66.0·11-s + (251. + 145. i)13-s + (−203. − 353. i)17-s + (−248. − 261. i)19-s + (−62.6 + 108. i)23-s + (−36.3 + 63.0i)25-s + (1.37e3 + 791. i)29-s + 457. i·31-s + (−52.1 − 90.3i)35-s + 2.19e3i·37-s + (1.95e3 − 1.13e3i)41-s + (1.26e3 + 2.18e3i)43-s + (1.32e3 − 2.29e3i)47-s + ⋯
L(s)  = 1  + (−0.528 − 0.915i)5-s + 0.0806·7-s + 0.545·11-s + (1.49 + 0.860i)13-s + (−0.705 − 1.22i)17-s + (−0.688 − 0.725i)19-s + (−0.118 + 0.205i)23-s + (−0.0581 + 0.100i)25-s + (1.63 + 0.941i)29-s + 0.476i·31-s + (−0.0426 − 0.0737i)35-s + 1.60i·37-s + (1.16 − 0.672i)41-s + (0.682 + 1.18i)43-s + (0.598 − 1.03i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.136 + 0.990i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.136 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.136 + 0.990i$
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: \((2,\ 684,\ (\ :2),\ 0.136 + 0.990i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.865686910\)
\(L(\frac12)\) \(\approx\) \(1.865686910\)
\(L(3)\) 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 \( 1 + (248. + 261. i)T \)
good5 \( 1 + (13.2 + 22.8i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 - 3.95T + 2.40e3T^{2} \)
11 \( 1 - 66.0T + 1.46e4T^{2} \)
13 \( 1 + (-251. - 145. i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (203. + 353. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (62.6 - 108. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-1.37e3 - 791. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 - 457. iT - 9.23e5T^{2} \)
37 \( 1 - 2.19e3iT - 1.87e6T^{2} \)
41 \( 1 + (-1.95e3 + 1.13e3i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-1.26e3 - 2.18e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-1.32e3 + 2.29e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (3.85e3 + 2.22e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-3.68e3 + 2.12e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-123. + 214. i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (6.64e3 + 3.83e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (2.60e3 - 1.50e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (1.79e3 + 3.11e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-6.43e3 + 3.71e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 1.65e3T + 4.74e7T^{2} \)
89 \( 1 + (-1.85e3 - 1.06e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (-1.27e4 + 7.36e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−9.410520291430352168596471116551, −8.788379651211094902321754502908, −8.249066448507185054982467216354, −6.91437933325953846593955979458, −6.29671134851807093880951867630, −4.81117901709585479879222729308, −4.37107071054397151416903959545, −3.10625728376377850526002041825, −1.55883924006938151786903808695, −0.54597342150037042989369405039, 1.02555044559592104367481121784, 2.45104353381199983234622895990, 3.66564316111193394273704292221, 4.24652841866998745992491202363, 6.04256324435140748199117113276, 6.29122990300194188087649775231, 7.58442206686356826311182822201, 8.281471336233124894217237398041, 9.102389557245190120005204634679, 10.56019866294526656494776306122

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