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Properties

Label 2-684-19.7-c5-0-38
Degree $2$
Conductor $684$
Sign $-0.463 + 0.886i$
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  + (18.6 + 32.3i)5-s + 15.8·7-s + 325.·11-s + (519. − 900. i)13-s + (−778. − 1.34e3i)17-s + (418. − 1.51e3i)19-s + (−784. + 1.35e3i)23-s + (866. − 1.50e3i)25-s + (−4.02e3 + 6.96e3i)29-s − 9.52e3·31-s + (296. + 512. i)35-s − 451.·37-s + (−2.23e3 − 3.86e3i)41-s + (5.77e3 + 1.00e4i)43-s + (3.08e3 − 5.33e3i)47-s + ⋯
L(s)  = 1  + (0.333 + 0.577i)5-s + 0.122·7-s + 0.811·11-s + (0.852 − 1.47i)13-s + (−0.653 − 1.13i)17-s + (0.265 − 0.964i)19-s + (−0.309 + 0.535i)23-s + (0.277 − 0.480i)25-s + (−0.888 + 1.53i)29-s − 1.78·31-s + (0.0408 + 0.0707i)35-s − 0.0541·37-s + (−0.207 − 0.359i)41-s + (0.476 + 0.825i)43-s + (0.203 − 0.352i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.463 + 0.886i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.463 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.463 + 0.886i$
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: \((2,\ 684,\ (\ :5/2),\ -0.463 + 0.886i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.427717442\)
\(L(\frac12)\) \(\approx\) \(1.427717442\)
\(L(\frac{7}{2})\) 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 \( 1 + (-418. + 1.51e3i)T \)
good5 \( 1 + (-18.6 - 32.3i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 - 15.8T + 1.68e4T^{2} \)
11 \( 1 - 325.T + 1.61e5T^{2} \)
13 \( 1 + (-519. + 900. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (778. + 1.34e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
23 \( 1 + (784. - 1.35e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (4.02e3 - 6.96e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + 9.52e3T + 2.86e7T^{2} \)
37 \( 1 + 451.T + 6.93e7T^{2} \)
41 \( 1 + (2.23e3 + 3.86e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (-5.77e3 - 1.00e4i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (-3.08e3 + 5.33e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-1.11e4 + 1.93e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-3.51e3 - 6.08e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-4.07e3 + 7.05e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (2.73e4 - 4.73e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (2.48e4 + 4.30e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 + (2.91e4 + 5.05e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (1.83e4 + 3.17e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 6.56e4T + 3.93e9T^{2} \)
89 \( 1 + (-2.01e4 + 3.48e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + (5.67e3 + 9.83e3i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−9.318861140878816739368638906611, −8.760926842262328381091246645979, −7.51482422033281006294372025817, −6.85569587385684886429358283371, −5.84170707665249474093487221798, −5.00224782796589021486423188943, −3.64389906523761426337803137193, −2.83548012526318646823366730131, −1.53145108463591887730801209550, −0.28678501095084183940752520143, 1.37386327254260054192802013231, 1.95324434654883780879436568191, 3.80324568006243948106235329875, 4.27580507584728981939628489510, 5.69242808355338692825189044031, 6.30927033806424825088162997188, 7.34695148068706150400188363979, 8.532630877996034807574961090704, 9.037617708516053039095907579269, 9.829311242109648292568331277945

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