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Properties

Label 2-684-19.7-c5-0-35
Degree $2$
Conductor $684$
Sign $-0.430 + 0.902i$
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  + (−35.6 − 61.7i)5-s + 252.·7-s − 88.0·11-s + (307. − 533. i)13-s + (−285. − 494. i)17-s + (361. − 1.53e3i)19-s + (1.21e3 − 2.10e3i)23-s + (−977. + 1.69e3i)25-s + (−1.14e3 + 1.97e3i)29-s + 3.68e3·31-s + (−8.99e3 − 1.55e4i)35-s + 3.06e3·37-s + (1.24e3 + 2.15e3i)41-s + (−2.45e3 − 4.24e3i)43-s + (−8.78e3 + 1.52e4i)47-s + ⋯
L(s)  = 1  + (−0.637 − 1.10i)5-s + 1.94·7-s − 0.219·11-s + (0.505 − 0.875i)13-s + (−0.239 − 0.415i)17-s + (0.229 − 0.973i)19-s + (0.478 − 0.829i)23-s + (−0.312 + 0.542i)25-s + (−0.252 + 0.437i)29-s + 0.688·31-s + (−1.24 − 2.14i)35-s + 0.367·37-s + (0.115 + 0.200i)41-s + (−0.202 − 0.350i)43-s + (−0.580 + 1.00i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.430 + 0.902i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.430 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.430 + 0.902i$
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: \((2,\ 684,\ (\ :5/2),\ -0.430 + 0.902i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.434027264\)
\(L(\frac12)\) \(\approx\) \(2.434027264\)
\(L(\frac{7}{2})\) 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 \( 1 + (-361. + 1.53e3i)T \)
good5 \( 1 + (35.6 + 61.7i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 - 252.T + 1.68e4T^{2} \)
11 \( 1 + 88.0T + 1.61e5T^{2} \)
13 \( 1 + (-307. + 533. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (285. + 494. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
23 \( 1 + (-1.21e3 + 2.10e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (1.14e3 - 1.97e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 - 3.68e3T + 2.86e7T^{2} \)
37 \( 1 - 3.06e3T + 6.93e7T^{2} \)
41 \( 1 + (-1.24e3 - 2.15e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (2.45e3 + 4.24e3i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (8.78e3 - 1.52e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-1.27e4 + 2.20e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-1.17e4 - 2.03e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-9.88e3 + 1.71e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (1.36e4 - 2.37e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (-1.67e4 - 2.90e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 + (8.98e3 + 1.55e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-4.12e4 - 7.14e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 4.02e4T + 3.93e9T^{2} \)
89 \( 1 + (-2.76e4 + 4.78e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + (-1.33e4 - 2.31e4i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−9.150526644159843937369690882926, −8.336625627420363781191602376974, −8.065100202232167261642306755421, −7.00074123225522124669180443933, −5.43687618579041961017100839171, −4.87530640710001814501550674014, −4.18815589447749064940255317543, −2.64074645211707319484672922921, −1.28422088302341203698529777587, −0.57233114429410498107880755086, 1.27864035255247311767454092502, 2.20958286942037452613025215220, 3.57971319954604338394606386321, 4.40384146493590358436201189867, 5.45293786588223834330021127186, 6.57747505541558873726152067785, 7.57845440514605078130643573890, 8.030069900608838064800832589753, 8.966204491486883207478654674716, 10.25480567936448777312008825834

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