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Properties

Label 2-684-171.103-c2-0-4
Degree $2$
Conductor $684$
Sign $0.0906 - 0.995i$
Analytic cond. $18.6376$
Root an. cond. $4.31713$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.40 + 2.65i)3-s + (−1.52 − 2.64i)5-s + (−5.58 − 9.67i)7-s + (−5.05 − 7.44i)9-s + (−3.62 − 6.28i)11-s + 10.5i·13-s + (9.16 − 0.334i)15-s + (−9.80 + 16.9i)17-s + (−8.02 + 17.2i)19-s + (33.4 − 1.22i)21-s + 12.2·23-s + (7.82 − 13.5i)25-s + (26.8 − 2.95i)27-s + (29.6 + 17.1i)29-s + (21.3 + 12.3i)31-s + ⋯
L(s)  = 1  + (−0.468 + 0.883i)3-s + (−0.305 − 0.529i)5-s + (−0.798 − 1.38i)7-s + (−0.561 − 0.827i)9-s + (−0.329 − 0.571i)11-s + 0.809i·13-s + (0.611 − 0.0223i)15-s + (−0.576 + 0.998i)17-s + (−0.422 + 0.906i)19-s + (1.59 − 0.0582i)21-s + 0.532·23-s + (0.313 − 0.542i)25-s + (0.994 − 0.109i)27-s + (1.02 + 0.591i)29-s + (0.687 + 0.397i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0906 - 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0906 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.0906 - 0.995i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (445, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ 0.0906 - 0.995i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7794694207\)
\(L(\frac12)\) \(\approx\) \(0.7794694207\)
\(L(2)\) 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 \( 1 + (1.40 - 2.65i)T \)
19 \( 1 + (8.02 - 17.2i)T \)
good5 \( 1 + (1.52 + 2.64i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (5.58 + 9.67i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (3.62 + 6.28i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 10.5iT - 169T^{2} \)
17 \( 1 + (9.80 - 16.9i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 - 12.2T + 529T^{2} \)
29 \( 1 + (-29.6 - 17.1i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-21.3 - 12.3i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 16.0iT - 1.36e3T^{2} \)
41 \( 1 + (5.22 - 3.01i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 - 69.9T + 1.84e3T^{2} \)
47 \( 1 + (31.7 - 55.0i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (54.5 - 31.4i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-61.1 + 35.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (49.8 - 86.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + 127. iT - 4.48e3T^{2} \)
71 \( 1 + (-5.62 - 3.24i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-10.1 + 17.5i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + 89.4iT - 6.24e3T^{2} \)
83 \( 1 + (-66.2 - 114. i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (125. - 72.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 56.2iT - 9.40e3T^{2} \)
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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

−10.62944041221490675778528758716, −9.758675887077008457290341887503, −8.855654356166815939459537713277, −8.026565299899077867770323929672, −6.68802198842262952396225933895, −6.13350416558193232788202876488, −4.68257264598243876534074845327, −4.14919268689305254281014988434, −3.17943846298160287455327992356, −0.961867912656532927833457805961, 0.37570734664142842085050694228, 2.41000280382174434828334226485, 2.91492625212037577523881732744, 4.82071987916615065288073262265, 5.69009145444876874294636644778, 6.61984953599140715100243558450, 7.22368874133911766609643978320, 8.285037942990243225583613091673, 9.134456777879728585310017716083, 10.13055172121275255618015727848

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