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Properties

Label 2-684-19.12-c4-0-20
Degree $2$
Conductor $684$
Sign $0.976 - 0.216i$
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  + (8.78 + 15.2i)5-s + 69.3·7-s − 198.·11-s + (215. + 124. i)13-s + (−112. − 194. i)17-s + (240. − 268. i)19-s + (462. − 800. i)23-s + (158. − 274. i)25-s + (964. + 557. i)29-s + 1.35e3i·31-s + (609. + 1.05e3i)35-s + 917. i·37-s + (−620. + 358. i)41-s + (−806. − 1.39e3i)43-s + (1.96e3 − 3.40e3i)47-s + ⋯
L(s)  = 1  + (0.351 + 0.608i)5-s + 1.41·7-s − 1.64·11-s + (1.27 + 0.734i)13-s + (−0.388 − 0.672i)17-s + (0.667 − 0.744i)19-s + (0.874 − 1.51i)23-s + (0.253 − 0.438i)25-s + (1.14 + 0.662i)29-s + 1.41i·31-s + (0.497 + 0.861i)35-s + 0.670i·37-s + (−0.369 + 0.213i)41-s + (−0.436 − 0.755i)43-s + (0.890 − 1.54i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.216i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.976 - 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.976 - 0.216i$
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: \((2,\ 684,\ (\ :2),\ 0.976 - 0.216i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.809601941\)
\(L(\frac12)\) \(\approx\) \(2.809601941\)
\(L(3)\) 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 \( 1 + (-240. + 268. i)T \)
good5 \( 1 + (-8.78 - 15.2i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 - 69.3T + 2.40e3T^{2} \)
11 \( 1 + 198.T + 1.46e4T^{2} \)
13 \( 1 + (-215. - 124. i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (112. + 194. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (-462. + 800. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-964. - 557. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 - 1.35e3iT - 9.23e5T^{2} \)
37 \( 1 - 917. iT - 1.87e6T^{2} \)
41 \( 1 + (620. - 358. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (806. + 1.39e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-1.96e3 + 3.40e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (1.55e3 + 900. i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (3.31e3 - 1.91e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-239. + 414. i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-3.19e3 - 1.84e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (-4.28e3 + 2.47e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (-1.55e3 - 2.69e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (4.95e3 - 2.86e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + 6.05e3T + 4.74e7T^{2} \)
89 \( 1 + (-1.19e4 - 6.92e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (-1.50e4 + 8.70e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−10.25130523770095348213306359618, −8.711858803060596902090474781612, −8.436708301531849006924551131167, −7.19732855773882557415535707120, −6.53346927838666686746403819584, −5.11852216105369670348247584720, −4.75006263441132221751021954467, −3.08164581517800500457994558869, −2.20371091581112106889121208659, −0.893645196594658069479904359225, 0.926454630605365530484408926768, 1.82937637983180810522321886522, 3.19722365698257865450818262556, 4.54026282152242946829257294611, 5.37954387961605402947977776436, 5.94535610861757243184503750988, 7.73694567936053900155902170495, 7.941959483713833310374088120194, 8.867447035932696798734898797833, 9.905881953836428008441553188935

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