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Properties

Label 2-684-171.103-c2-0-27
Degree $2$
Conductor $684$
Sign $0.767 + 0.641i$
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  + (−0.455 + 2.96i)3-s + (1.91 + 3.31i)5-s + (−3.97 − 6.88i)7-s + (−8.58 − 2.70i)9-s + (2.64 + 4.58i)11-s − 14.2i·13-s + (−10.6 + 4.16i)15-s + (6.41 − 11.1i)17-s + (2.65 − 18.8i)19-s + (22.2 − 8.64i)21-s − 18.5·23-s + (5.18 − 8.98i)25-s + (11.9 − 24.2i)27-s + (24.1 + 13.9i)29-s + (−22.1 − 12.7i)31-s + ⋯
L(s)  = 1  + (−0.151 + 0.988i)3-s + (0.382 + 0.662i)5-s + (−0.567 − 0.983i)7-s + (−0.953 − 0.300i)9-s + (0.240 + 0.416i)11-s − 1.09i·13-s + (−0.712 + 0.277i)15-s + (0.377 − 0.653i)17-s + (0.139 − 0.990i)19-s + (1.05 − 0.411i)21-s − 0.807·23-s + (0.207 − 0.359i)25-s + (0.441 − 0.897i)27-s + (0.833 + 0.480i)29-s + (−0.714 − 0.412i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 + 0.641i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.767 + 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.767 + 0.641i$
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: \((2,\ 684,\ (\ :1),\ 0.767 + 0.641i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.265292607\)
\(L(\frac12)\) \(\approx\) \(1.265292607\)
\(L(2)\) 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 \( 1 + (0.455 - 2.96i)T \)
19 \( 1 + (-2.65 + 18.8i)T \)
good5 \( 1 + (-1.91 - 3.31i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (3.97 + 6.88i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-2.64 - 4.58i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 14.2iT - 169T^{2} \)
17 \( 1 + (-6.41 + 11.1i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + 18.5T + 529T^{2} \)
29 \( 1 + (-24.1 - 13.9i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (22.1 + 12.7i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 12.6iT - 1.36e3T^{2} \)
41 \( 1 + (29.4 - 16.9i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 - 26.6T + 1.84e3T^{2} \)
47 \( 1 + (5.03 - 8.71i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-26.8 + 15.5i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-23.0 + 13.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-14.9 + 25.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 - 48.5iT - 4.48e3T^{2} \)
71 \( 1 + (-38.6 - 22.3i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (18.2 - 31.6i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + 2.68iT - 6.24e3T^{2} \)
83 \( 1 + (65.5 + 113. i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-133. + 76.9i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 101. iT - 9.40e3T^{2} \)
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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

−10.12504704693375667989432133674, −9.716699350092377003664189760530, −8.612428290971712311483968137489, −7.42038314090194782863498768540, −6.62035965236647875462562386735, −5.61916557035401547842968867094, −4.59797852452806048923764633606, −3.54678461349961533171915698657, −2.69550942174611484608224716071, −0.48494628945584979356407961745, 1.31804326569949127297317061223, 2.30961723355402671766484139816, 3.70012617180804945256484321138, 5.24652417937095607027208611595, 5.99410739027371767187735402359, 6.62188291331311934033397900264, 7.85076045586484744788668690085, 8.699087115758617062985547487337, 9.252088415387417246904028122972, 10.32756709234551869392679575444

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