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Properties

Label 2-684-171.103-c2-0-21
Degree $2$
Conductor $684$
Sign $0.694 + 0.719i$
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  + (−0.224 − 2.99i)3-s + (1.33 + 2.31i)5-s + (3.94 + 6.83i)7-s + (−8.89 + 1.34i)9-s + (−4.28 − 7.41i)11-s − 9.02i·13-s + (6.63 − 4.52i)15-s + (−5.47 + 9.47i)17-s + (12.2 − 14.5i)19-s + (19.5 − 13.3i)21-s + 30.7·23-s + (8.91 − 15.4i)25-s + (6.01 + 26.3i)27-s + (33.2 + 19.2i)29-s + (4.68 + 2.70i)31-s + ⋯
L(s)  = 1  + (−0.0747 − 0.997i)3-s + (0.267 + 0.463i)5-s + (0.564 + 0.977i)7-s + (−0.988 + 0.149i)9-s + (−0.389 − 0.674i)11-s − 0.694i·13-s + (0.442 − 0.301i)15-s + (−0.321 + 0.557i)17-s + (0.645 − 0.763i)19-s + (0.932 − 0.635i)21-s + 1.33·23-s + (0.356 − 0.617i)25-s + (0.222 + 0.974i)27-s + (1.14 + 0.662i)29-s + (0.151 + 0.0872i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.694 + 0.719i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.694 + 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.694 + 0.719i$
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: \((2,\ 684,\ (\ :1),\ 0.694 + 0.719i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.915004780\)
\(L(\frac12)\) \(\approx\) \(1.915004780\)
\(L(2)\) 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 \( 1 + (0.224 + 2.99i)T \)
19 \( 1 + (-12.2 + 14.5i)T \)
good5 \( 1 + (-1.33 - 2.31i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-3.94 - 6.83i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (4.28 + 7.41i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 9.02iT - 169T^{2} \)
17 \( 1 + (5.47 - 9.47i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 - 30.7T + 529T^{2} \)
29 \( 1 + (-33.2 - 19.2i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-4.68 - 2.70i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 5.16iT - 1.36e3T^{2} \)
41 \( 1 + (-0.170 + 0.0984i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 - 57.5T + 1.84e3T^{2} \)
47 \( 1 + (-26.2 + 45.3i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-47.6 + 27.4i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (34.5 - 19.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-21.4 + 37.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 - 93.6iT - 4.48e3T^{2} \)
71 \( 1 + (96.9 + 55.9i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-59.8 + 103. i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + 22.7iT - 6.24e3T^{2} \)
83 \( 1 + (-4.68 - 8.12i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (127. - 73.5i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 122. iT - 9.40e3T^{2} \)
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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

−10.43940134047774370246316967827, −8.912022261226323920092612473133, −8.522453216588743784180618829360, −7.50672198625286895294582861208, −6.62140226606348861708065975883, −5.72752014876913480146228875962, −5.01585748019178584906013202970, −3.05198493302592943756928438249, −2.38823707900978487534195797062, −0.882067865853326543242569142070, 1.09130713742292746937095441811, 2.76710426998146257695469974649, 4.17797620915880016822324369583, 4.71673872819084635007455752969, 5.61315785163282022928051596804, 6.94288262393091332003106306186, 7.80807981786390810234565391894, 8.908609360624322793730456771792, 9.553742625955244463506651449732, 10.36928189774555720939153061026

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