Properties

Label 2-2646-63.59-c1-0-30
Degree $2$
Conductor $2646$
Sign $0.452 + 0.891i$
Analytic cond. $21.1284$
Root an. cond. $4.59656$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + 3.89·5-s − 0.999i·8-s + (−3.36 − 1.94i)10-s − 3.94i·11-s + (2.46 + 1.42i)13-s + (−0.5 + 0.866i)16-s + (0.371 − 0.642i)17-s + (−1.54 + 0.892i)19-s + (1.94 + 3.36i)20-s + (−1.97 + 3.41i)22-s − 6.25i·23-s + 10.1·25-s + (−1.42 − 2.46i)26-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + 1.74·5-s − 0.353i·8-s + (−1.06 − 0.615i)10-s − 1.18i·11-s + (0.684 + 0.395i)13-s + (−0.125 + 0.216i)16-s + (0.0899 − 0.155i)17-s + (−0.354 + 0.204i)19-s + (0.435 + 0.753i)20-s + (−0.420 + 0.728i)22-s − 1.30i·23-s + 2.02·25-s + (−0.279 − 0.483i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.452 + 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.452 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.452 + 0.891i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (2285, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 0.452 + 0.891i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.936322365\)
\(L(\frac12)\) \(\approx\) \(1.936322365\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 3.89T + 5T^{2} \)
11 \( 1 + 3.94iT - 11T^{2} \)
13 \( 1 + (-2.46 - 1.42i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.371 + 0.642i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.54 - 0.892i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 6.25iT - 23T^{2} \)
29 \( 1 + (-2.50 + 1.44i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.04 - 1.75i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.50 + 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.24 + 9.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.471 - 0.816i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.09 - 1.89i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.0105 + 0.0183i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.13 - 1.23i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.72 + 11.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.94iT - 71T^{2} \)
73 \( 1 + (4.20 + 2.42i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.81 - 3.14i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.02 - 6.98i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.63 + 8.02i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-16.2 + 9.40i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.899985990178280173344107143589, −8.320557959603371068244547925719, −7.16389346119183178860358057623, −6.20292202196322995918726396796, −5.98089254081077349473383054684, −4.89691541265035422942674202112, −3.68581339740307958333332143552, −2.66514019976666613788379964967, −1.89858886579751851945097530230, −0.822821105935683801300513620915, 1.31193132257066979281342803834, 1.98505041682540053778321448037, 3.02389206454463496470898039213, 4.46894029643891157811209139769, 5.40488662847417654745005540304, 5.94800049423874703560853248000, 6.69261515115478587822429676228, 7.40630440162840917049432950556, 8.345786725413239434302239792433, 9.152408421795020274813558105588

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