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Properties

Label 2-684-171.103-c2-0-37
Degree $2$
Conductor $684$
Sign $-0.348 - 0.937i$
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  + (−2.21 − 2.01i)3-s + (−2.05 − 3.55i)5-s + (−6.48 − 11.2i)7-s + (0.840 + 8.96i)9-s + (−7.93 − 13.7i)11-s + 0.390i·13-s + (−2.62 + 12.0i)15-s + (7.48 − 12.9i)17-s + (−9.33 − 16.5i)19-s + (−8.30 + 38.0i)21-s − 18.6·23-s + (4.07 − 7.05i)25-s + (16.2 − 21.5i)27-s + (13.2 + 7.62i)29-s + (12.7 + 7.34i)31-s + ⋯
L(s)  = 1  + (−0.739 − 0.673i)3-s + (−0.410 − 0.711i)5-s + (−0.927 − 1.60i)7-s + (0.0933 + 0.995i)9-s + (−0.721 − 1.25i)11-s + 0.0300i·13-s + (−0.175 + 0.802i)15-s + (0.440 − 0.762i)17-s + (−0.491 − 0.870i)19-s + (−0.395 + 1.81i)21-s − 0.812·23-s + (0.162 − 0.282i)25-s + (0.601 − 0.798i)27-s + (0.455 + 0.262i)29-s + (0.410 + 0.236i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.348 - 0.937i$
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: \((2,\ 684,\ (\ :1),\ -0.348 - 0.937i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5550441133\)
\(L(\frac12)\) \(\approx\) \(0.5550441133\)
\(L(2)\) 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 \( 1 + (2.21 + 2.01i)T \)
19 \( 1 + (9.33 + 16.5i)T \)
good5 \( 1 + (2.05 + 3.55i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (6.48 + 11.2i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (7.93 + 13.7i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 0.390iT - 169T^{2} \)
17 \( 1 + (-7.48 + 12.9i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + 18.6T + 529T^{2} \)
29 \( 1 + (-13.2 - 7.62i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-12.7 - 7.34i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 35.7iT - 1.36e3T^{2} \)
41 \( 1 + (-19.8 + 11.4i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 - 16.0T + 1.84e3T^{2} \)
47 \( 1 + (-21.4 + 37.2i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-17.6 + 10.2i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (60.1 - 34.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-38.0 + 65.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 - 44.8iT - 4.48e3T^{2} \)
71 \( 1 + (-20.3 - 11.7i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-33.0 + 57.2i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 - 120. iT - 6.24e3T^{2} \)
83 \( 1 + (65.1 + 112. i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (113. - 65.5i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 144. iT - 9.40e3T^{2} \)
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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

−9.936584260334977724106544132622, −8.602125532006163241274831179476, −7.81801373205394841605894004692, −6.99199029583565726762126171440, −6.22550144402368708566068431969, −5.10910319133515714701851195061, −4.16750443352472724546245672471, −2.90347166411892078162443778785, −0.873171647940303997744416427988, −0.29643534811895484353863292158, 2.27813798824172773889316241722, 3.37697659632806121357980795229, 4.46695723296394439171538673651, 5.71812681228435719434370785934, 6.13699574583058351627596983179, 7.25149342964957014647701695715, 8.356192282300478350572463124040, 9.449907409402238431117224480498, 10.03090861556051980957590244325, 10.74083747728017298111475083981

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