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Properties

Label 2-684-19.11-c5-0-38
Degree $2$
Conductor $684$
Sign $-0.214 + 0.976i$
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  + (47.4 − 82.2i)5-s + 189.·7-s + 530.·11-s + (−353. − 612. i)13-s + (764. − 1.32e3i)17-s + (654. − 1.43e3i)19-s + (−497. − 861. i)23-s + (−2.94e3 − 5.10e3i)25-s + (1.29e3 + 2.23e3i)29-s − 2.79e3·31-s + (9.01e3 − 1.56e4i)35-s + 7.23e3·37-s + (−2.18e3 + 3.77e3i)41-s + (−3.12e3 + 5.40e3i)43-s + (−1.23e4 − 2.13e4i)47-s + ⋯
L(s)  = 1  + (0.849 − 1.47i)5-s + 1.46·7-s + 1.32·11-s + (−0.580 − 1.00i)13-s + (0.641 − 1.11i)17-s + (0.415 − 0.909i)19-s + (−0.196 − 0.339i)23-s + (−0.943 − 1.63i)25-s + (0.284 + 0.493i)29-s − 0.521·31-s + (1.24 − 2.15i)35-s + 0.869·37-s + (−0.202 + 0.351i)41-s + (−0.257 + 0.445i)43-s + (−0.813 − 1.40i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.214 + 0.976i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.214 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.214 + 0.976i$
Analytic conductor: \(109.702\)
Root analytic conductor: \(10.4738\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (505, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :5/2),\ -0.214 + 0.976i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.685468983\)
\(L(\frac12)\) \(\approx\) \(3.685468983\)
\(L(\frac{7}{2})\) 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 \( 1 + (-654. + 1.43e3i)T \)
good5 \( 1 + (-47.4 + 82.2i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 - 189.T + 1.68e4T^{2} \)
11 \( 1 - 530.T + 1.61e5T^{2} \)
13 \( 1 + (353. + 612. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (-764. + 1.32e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
23 \( 1 + (497. + 861. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (-1.29e3 - 2.23e3i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + 2.79e3T + 2.86e7T^{2} \)
37 \( 1 - 7.23e3T + 6.93e7T^{2} \)
41 \( 1 + (2.18e3 - 3.77e3i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (3.12e3 - 5.40e3i)T + (-7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + (1.23e4 + 2.13e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-1.54e4 - 2.68e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-1.81e4 + 3.14e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-2.88e3 - 5.00e3i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-3.03e4 - 5.26e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (1.48e4 - 2.56e4i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 + (3.90e4 - 6.75e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (3.34e4 - 5.78e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 2.19e4T + 3.93e9T^{2} \)
89 \( 1 + (-4.53e4 - 7.85e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + (3.07e4 - 5.32e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−9.389161501126694321541226248089, −8.662515021688905762137241257188, −7.937162115579723109011543806561, −6.86137536222830482460724880468, −5.40731728864796925264780071570, −5.14645454689593130229822592844, −4.20170018929066972394756808067, −2.53597002001404541343845478287, −1.31885320887556455323081311591, −0.815744886970347930006481232361, 1.54710663439217735863219270035, 1.97325616357226237304090279028, 3.41976001247719492298343638860, 4.38777001092229990827041552928, 5.68434164425938422133999472078, 6.39627690120702965574814482763, 7.28437006788105488837480646315, 8.104474742678596341240266451221, 9.270755005586392383106373521998, 10.00854369328378492884695847653

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