Properties

Label 2-684-19.12-c4-0-12
Degree $2$
Conductor $684$
Sign $0.652 - 0.757i$
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  + (8.82 + 15.2i)5-s − 52.3·7-s + 204.·11-s + (278. + 160. i)13-s + (−194. − 337. i)17-s + (−32.1 + 359. i)19-s + (88.9 − 154. i)23-s + (156. − 271. i)25-s + (−985. − 569. i)29-s − 263. i·31-s + (−461. − 799. i)35-s + 702. i·37-s + (1.38e3 − 801. i)41-s + (28.5 + 49.5i)43-s + (−533. + 924. i)47-s + ⋯
L(s)  = 1  + (0.352 + 0.611i)5-s − 1.06·7-s + 1.69·11-s + (1.64 + 0.952i)13-s + (−0.673 − 1.16i)17-s + (−0.0891 + 0.996i)19-s + (0.168 − 0.291i)23-s + (0.251 − 0.434i)25-s + (−1.17 − 0.676i)29-s − 0.274i·31-s + (−0.377 − 0.653i)35-s + 0.513i·37-s + (0.826 − 0.477i)41-s + (0.0154 + 0.0267i)43-s + (−0.241 + 0.418i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.652 - 0.757i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.652 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.652 - 0.757i$
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: \((2,\ 684,\ (\ :2),\ 0.652 - 0.757i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.321456933\)
\(L(\frac12)\) \(\approx\) \(2.321456933\)
\(L(3)\) 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 \( 1 + (32.1 - 359. i)T \)
good5 \( 1 + (-8.82 - 15.2i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + 52.3T + 2.40e3T^{2} \)
11 \( 1 - 204.T + 1.46e4T^{2} \)
13 \( 1 + (-278. - 160. i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (194. + 337. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (-88.9 + 154. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (985. + 569. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + 263. iT - 9.23e5T^{2} \)
37 \( 1 - 702. iT - 1.87e6T^{2} \)
41 \( 1 + (-1.38e3 + 801. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-28.5 - 49.5i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (533. - 924. i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (-2.23e3 - 1.28e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-2.52e3 + 1.45e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (2.03e3 - 3.51e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-6.20e3 - 3.58e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (-7.39e3 + 4.27e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (-4.14e3 - 7.17e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (3.14e3 - 1.81e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 1.16e3T + 4.74e7T^{2} \)
89 \( 1 + (-3.70e3 - 2.13e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (6.20e3 - 3.58e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−9.781024655582985433430291128056, −9.309933815675535910570538240331, −8.502410338733801956572678568806, −7.00419948828303076334104152302, −6.49254305873921561295153532837, −5.90750006665426725387876919334, −4.16014988000769984291573921358, −3.57333783614841978179842312970, −2.26098191826769926255313327511, −0.974891452671369386920778879367, 0.70184408463091140236405520525, 1.68297313825089981672998956947, 3.37694579344183948272696603134, 3.97120452696774769205085110588, 5.41003918687617475196144014327, 6.28564200961153539711175315305, 6.83887110137228145002999997217, 8.306875400879807519080204834482, 9.074596743176749615894755114738, 9.463139637964229577340204865952

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