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Properties

Label 2-684-19.12-c4-0-7
Degree $2$
Conductor $684$
Sign $-0.574 - 0.818i$
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 + 22.0i)5-s + 27.7·7-s − 143.·11-s + (123. + 71.5i)13-s + (−43.9 − 76.1i)17-s + (340. + 119. i)19-s + (−350. + 607. i)23-s + (−12.6 + 21.9i)25-s + (37.5 + 21.6i)29-s − 111. i·31-s + (353. + 613. i)35-s + 1.07e3i·37-s + (−80.1 + 46.2i)41-s + (945. + 1.63e3i)43-s + (−1.36e3 + 2.36e3i)47-s + ⋯
L(s)  = 1  + (0.510 + 0.883i)5-s + 0.566·7-s − 1.18·11-s + (0.733 + 0.423i)13-s + (−0.152 − 0.263i)17-s + (0.943 + 0.331i)19-s + (−0.663 + 1.14i)23-s + (−0.0202 + 0.0350i)25-s + (0.0446 + 0.0257i)29-s − 0.115i·31-s + (0.288 + 0.500i)35-s + 0.788i·37-s + (−0.0476 + 0.0275i)41-s + (0.511 + 0.886i)43-s + (−0.618 + 1.07i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.574 - 0.818i$
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: \((2,\ 684,\ (\ :2),\ -0.574 - 0.818i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.781176104\)
\(L(\frac12)\) \(\approx\) \(1.781176104\)
\(L(3)\) 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 \( 1 + (-340. - 119. i)T \)
good5 \( 1 + (-12.7 - 22.0i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 - 27.7T + 2.40e3T^{2} \)
11 \( 1 + 143.T + 1.46e4T^{2} \)
13 \( 1 + (-123. - 71.5i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (43.9 + 76.1i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (350. - 607. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-37.5 - 21.6i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + 111. iT - 9.23e5T^{2} \)
37 \( 1 - 1.07e3iT - 1.87e6T^{2} \)
41 \( 1 + (80.1 - 46.2i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-945. - 1.63e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (1.36e3 - 2.36e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (2.15e3 + 1.24e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-5.03e3 + 2.90e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-1.26e3 + 2.18e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-4.30e3 - 2.48e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (3.72e3 - 2.14e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (1.58e3 + 2.75e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-4.93e3 + 2.84e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + 1.04e4T + 4.74e7T^{2} \)
89 \( 1 + (4.82e3 + 2.78e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (1.56e4 - 9.01e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−10.11905399520323032666239803345, −9.584405259180387706202529862674, −8.281351984097270115006638337446, −7.65519660891544954758274907058, −6.62720627680609617507805624473, −5.74617998104662503214838301942, −4.84905709559018118830281695656, −3.49447338157512986877414552669, −2.51353628375482842797560438990, −1.38303164397992164385189357189, 0.41529331121491122978778448405, 1.56783175217704588937470899966, 2.73495413662357157668642117465, 4.14615178499326103091198333690, 5.22101875476634732232965637058, 5.67519341800030199395259544882, 6.99958791669773355595348002623, 8.124634053692347231120210351423, 8.563268705324058433588937615301, 9.594745262914273006801473222072

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