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Properties

Label 2-684-19.7-c5-0-19
Degree $2$
Conductor $684$
Sign $0.981 + 0.192i$
Analytic cond. $109.702$
Root an. cond. $10.4738$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (15.4 + 26.7i)5-s + 132.·7-s − 670.·11-s + (−411. + 712. i)13-s + (−731. − 1.26e3i)17-s + (−1.57e3 + 30.7i)19-s + (1.14e3 − 1.98e3i)23-s + (1.08e3 − 1.87e3i)25-s + (1.38e3 − 2.39e3i)29-s + 1.05e4·31-s + (2.04e3 + 3.54e3i)35-s + 4.81e3·37-s + (7.28e3 + 1.26e4i)41-s + (−5.15e3 − 8.92e3i)43-s + (2.55e3 − 4.42e3i)47-s + ⋯
L(s)  = 1  + (0.276 + 0.479i)5-s + 1.01·7-s − 1.67·11-s + (−0.675 + 1.16i)13-s + (−0.613 − 1.06i)17-s + (−0.999 + 0.0195i)19-s + (0.451 − 0.782i)23-s + (0.346 − 0.600i)25-s + (0.304 − 0.528i)29-s + 1.97·31-s + (0.282 + 0.488i)35-s + 0.578·37-s + (0.676 + 1.17i)41-s + (−0.424 − 0.735i)43-s + (0.168 − 0.292i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.981 + 0.192i$
Analytic conductor: \(109.702\)
Root analytic conductor: \(10.4738\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :5/2),\ 0.981 + 0.192i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.988545430\)
\(L(\frac12)\) \(\approx\) \(1.988545430\)
\(L(\frac{7}{2})\) 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 \( 1 \)
19 \( 1 + (1.57e3 - 30.7i)T \)
good5 \( 1 + (-15.4 - 26.7i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 - 132.T + 1.68e4T^{2} \)
11 \( 1 + 670.T + 1.61e5T^{2} \)
13 \( 1 + (411. - 712. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (731. + 1.26e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
23 \( 1 + (-1.14e3 + 1.98e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (-1.38e3 + 2.39e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 - 1.05e4T + 2.86e7T^{2} \)
37 \( 1 - 4.81e3T + 6.93e7T^{2} \)
41 \( 1 + (-7.28e3 - 1.26e4i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (5.15e3 + 8.92e3i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (-2.55e3 + 4.42e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-3.72e3 + 6.45e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-1.71e4 - 2.97e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-2.06e4 + 3.56e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (1.96e4 - 3.39e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (5.84e3 + 1.01e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 + (1.36e4 + 2.37e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (2.58e3 + 4.47e3i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 8.61e4T + 3.93e9T^{2} \)
89 \( 1 + (2.53e4 - 4.39e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + (1.33e4 + 2.30e4i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−9.869729944880828884761763002211, −8.680039197857019830948824894486, −8.004271811093274436982109613022, −7.04429560592908684881972786586, −6.24043451237583125694753935133, −4.80882398696817853157262584630, −4.59653882468930708341725453297, −2.60868642847493339457046670842, −2.27146072190488350137082275105, −0.56347423917340544494037020758, 0.74879863873150318290543859826, 1.99596939608398390256430592680, 2.94633387709459390500267977429, 4.53026620823884734656268992981, 5.13257046466861517939915640789, 5.95582341889489230405990158561, 7.33435338890084654022266930549, 8.144830511049808385418057924369, 8.594675272871714372123337599753, 9.907393477637659985342927240216

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