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Properties

Label 2-684-171.103-c2-0-7
Degree $2$
Conductor $684$
Sign $-0.680 - 0.732i$
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  + (−1.99 + 2.23i)3-s + (0.267 + 0.464i)5-s + (4.62 + 8.00i)7-s + (−1.01 − 8.94i)9-s + (6.99 + 12.1i)11-s − 1.06i·13-s + (−1.57 − 0.327i)15-s + (−0.480 + 0.832i)17-s + (14.7 + 12.0i)19-s + (−27.1 − 5.64i)21-s − 0.00791·23-s + (12.3 − 21.4i)25-s + (22.0 + 15.5i)27-s + (17.0 + 9.82i)29-s + (−0.693 − 0.400i)31-s + ⋯
L(s)  = 1  + (−0.665 + 0.746i)3-s + (0.0535 + 0.0928i)5-s + (0.660 + 1.14i)7-s + (−0.113 − 0.993i)9-s + (0.635 + 1.10i)11-s − 0.0819i·13-s + (−0.104 − 0.0218i)15-s + (−0.0282 + 0.0489i)17-s + (0.774 + 0.632i)19-s + (−1.29 − 0.268i)21-s − 0.000344·23-s + (0.494 − 0.856i)25-s + (0.816 + 0.577i)27-s + (0.586 + 0.338i)29-s + (−0.0223 − 0.0129i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.680 - 0.732i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.680 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.680 - 0.732i$
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: \((2,\ 684,\ (\ :1),\ -0.680 - 0.732i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.426953476\)
\(L(\frac12)\) \(\approx\) \(1.426953476\)
\(L(2)\) 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 \( 1 + (1.99 - 2.23i)T \)
19 \( 1 + (-14.7 - 12.0i)T \)
good5 \( 1 + (-0.267 - 0.464i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-4.62 - 8.00i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-6.99 - 12.1i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 1.06iT - 169T^{2} \)
17 \( 1 + (0.480 - 0.832i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + 0.00791T + 529T^{2} \)
29 \( 1 + (-17.0 - 9.82i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (0.693 + 0.400i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 14.3iT - 1.36e3T^{2} \)
41 \( 1 + (42.4 - 24.5i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + 15.1T + 1.84e3T^{2} \)
47 \( 1 + (5.03 - 8.72i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (40.3 - 23.2i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (65.5 - 37.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-5.92 + 10.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 - 25.7iT - 4.48e3T^{2} \)
71 \( 1 + (32.7 + 18.9i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (17.6 - 30.5i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 - 11.0iT - 6.24e3T^{2} \)
83 \( 1 + (27.9 + 48.3i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (42.2 - 24.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 92.5iT - 9.40e3T^{2} \)
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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

−10.51264429414995322208207493071, −9.794703191243245777913267421760, −9.032027789356911507558837940530, −8.153577772839230622143361030448, −6.88473077014933629724689226239, −6.00123317673639653230967423892, −5.07695648870768302924326877521, −4.38637983594703344081448218793, −3.01443981784776054152455869399, −1.53517456399813463805116234181, 0.61059672594400686163954866319, 1.54929999262962222774820076490, 3.27415896677510576975299299181, 4.56414975869672657719632108290, 5.45259517860957550266813734661, 6.52527068471192494746564400366, 7.23694156433088430608113107704, 8.052955021707013901611766384024, 8.975256157662325367344123148067, 10.18913476197902440343665959673

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