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Properties

Label 2-684-19.12-c4-0-18
Degree $2$
Conductor $684$
Sign $0.808 + 0.588i$
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  + (11.9 + 20.7i)5-s − 2.95·7-s − 190.·11-s + (−22.7 − 13.1i)13-s + (57.0 + 98.8i)17-s + (−360. − 2.68i)19-s + (496. − 860. i)23-s + (25.0 − 43.3i)25-s + (472. + 272. i)29-s − 512. i·31-s + (−35.4 − 61.4i)35-s − 459. i·37-s + (2.72e3 − 1.57e3i)41-s + (−551. − 955. i)43-s + (−526. + 911. i)47-s + ⋯
L(s)  = 1  + (0.479 + 0.830i)5-s − 0.0603·7-s − 1.57·11-s + (−0.134 − 0.0778i)13-s + (0.197 + 0.342i)17-s + (−0.999 − 0.00744i)19-s + (0.939 − 1.62i)23-s + (0.0400 − 0.0693i)25-s + (0.561 + 0.324i)29-s − 0.533i·31-s + (−0.0289 − 0.0501i)35-s − 0.335i·37-s + (1.61 − 0.935i)41-s + (−0.298 − 0.516i)43-s + (−0.238 + 0.412i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 + 0.588i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.808 + 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.808 + 0.588i$
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: \((2,\ 684,\ (\ :2),\ 0.808 + 0.588i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.635452540\)
\(L(\frac12)\) \(\approx\) \(1.635452540\)
\(L(3)\) 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 \( 1 + (360. + 2.68i)T \)
good5 \( 1 + (-11.9 - 20.7i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + 2.95T + 2.40e3T^{2} \)
11 \( 1 + 190.T + 1.46e4T^{2} \)
13 \( 1 + (22.7 + 13.1i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (-57.0 - 98.8i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (-496. + 860. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-472. - 272. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + 512. iT - 9.23e5T^{2} \)
37 \( 1 + 459. iT - 1.87e6T^{2} \)
41 \( 1 + (-2.72e3 + 1.57e3i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (551. + 955. i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (526. - 911. i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (-2.49e3 - 1.44e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-2.79e3 + 1.61e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (2.89e3 - 5.01e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-3.58e3 - 2.06e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (-2.38e3 + 1.37e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (-3.13e3 - 5.42e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (867. - 500. i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 1.98e3T + 4.74e7T^{2} \)
89 \( 1 + (-1.64e3 - 948. i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (-2.57e3 + 1.48e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−10.15863565988419536587383300645, −8.885314934667499681236053601900, −8.076607540031770792144558120636, −7.09688151780170003368351697785, −6.28975226433385205243597994061, −5.36174363193003838826016219928, −4.29522556073759106238643011822, −2.83178091269039939762269563750, −2.29948612461205570917579593570, −0.46861779071316355947277926423, 0.895730055324671595172462470309, 2.16044682300560819912538623467, 3.28377457256951484463382712098, 4.80967670959610906301820021489, 5.25302231755471777547698013987, 6.33068903396859384720965170974, 7.51411720516331982538786140305, 8.241059971019535827268078517239, 9.194259599741865296276172260353, 9.881882353887979410993141685334

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