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Properties

Label 2-2646-63.4-c1-0-6
Degree $2$
Conductor $2646$
Sign $-0.559 - 0.828i$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s − 4.37·5-s + 0.999·8-s + (2.18 − 3.78i)10-s + 1.37·11-s + (−1 + 1.73i)13-s + (−0.5 + 0.866i)16-s + (0.686 − 1.18i)17-s + (−2.5 − 4.33i)19-s + (2.18 + 3.78i)20-s + (−0.686 + 1.18i)22-s + 1.62·23-s + 14.1·25-s + (−0.999 − 1.73i)26-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s − 1.95·5-s + 0.353·8-s + (0.691 − 1.19i)10-s + 0.413·11-s + (−0.277 + 0.480i)13-s + (−0.125 + 0.216i)16-s + (0.166 − 0.288i)17-s + (−0.573 − 0.993i)19-s + (0.488 + 0.846i)20-s + (−0.146 + 0.253i)22-s + 0.339·23-s + 2.82·25-s + (−0.196 − 0.339i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.559 - 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.559 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.559 - 0.828i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ -0.559 - 0.828i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4578071674\)
\(L(\frac12)\) \(\approx\) \(0.4578071674\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 4.37T + 5T^{2} \)
11 \( 1 - 1.37T + 11T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.686 + 1.18i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.62T + 23T^{2} \)
29 \( 1 + (4.37 + 7.57i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.31 + 4.00i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.05 - 7.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.37 - 7.57i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.05 + 8.76i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.55 - 2.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.05 - 1.83i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.11T + 71T^{2} \)
73 \( 1 + (6.05 - 10.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.55 + 4.43i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.74 - 15.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.37 - 12.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.05 - 7.02i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.075236700587501965210457816008, −8.124808342405388511272713552785, −7.68206997588125667594110240582, −7.01299130388191165441014999316, −6.34596704540668470434794396383, −5.11020444346490057931708584325, −4.32830698603519802808455166401, −3.79336767974122541992016669289, −2.55381423426269280673360830379, −0.817645971777291304580223704905, 0.25346285820341548265606375768, 1.55152617679089887053512646638, 3.07596571046677670691874317417, 3.63116879773292896030957052502, 4.34871802716806614813430058164, 5.23127497932271184477592250270, 6.52909674291594173444756900864, 7.42828440304770580599152972627, 7.84744558500904813901744223875, 8.616851655551677360876459561435

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