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Properties

Label 2-2646-63.25-c1-0-36
Degree $2$
Conductor $2646$
Sign $-0.415 + 0.909i$
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  + 2-s + 4-s + (1.5 − 2.59i)5-s + 8-s + (1.5 − 2.59i)10-s + (−3 − 5.19i)11-s + (1 + 1.73i)13-s + 16-s + (−3 + 5.19i)17-s + (−3.5 − 6.06i)19-s + (1.5 − 2.59i)20-s + (−3 − 5.19i)22-s + (1.5 − 2.59i)23-s + (−2 − 3.46i)25-s + (1 + 1.73i)26-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (0.670 − 1.16i)5-s + 0.353·8-s + (0.474 − 0.821i)10-s + (−0.904 − 1.56i)11-s + (0.277 + 0.480i)13-s + 0.250·16-s + (−0.727 + 1.26i)17-s + (−0.802 − 1.39i)19-s + (0.335 − 0.580i)20-s + (−0.639 − 1.10i)22-s + (0.312 − 0.541i)23-s + (−0.400 − 0.692i)25-s + (0.196 + 0.339i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.415 + 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.415 + 0.909i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (2125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ -0.415 + 0.909i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.553209099\)
\(L(\frac12)\) \(\approx\) \(2.553209099\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1 - 1.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 5T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 5T + 79T^{2} \)
83 \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1 + 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.582802745365980176714549917598, −8.123063428493459659180928151609, −6.76896785249808213671159573099, −6.14115125480109365377394483201, −5.47657645947162229343466478161, −4.72150674143783035808953228403, −3.99476402641683646171195354192, −2.80735051271798484923763990792, −1.91200897076389879440871610987, −0.59350255717347315057394342410, 1.81132020339380924657786265295, 2.54395127821462212286025034497, 3.31419653659577975557109894294, 4.45053261262822048545312171887, 5.21566168260829610499324672835, 5.98342890635902755421339175316, 6.87844093293021943315136714684, 7.24043210112474355261780469862, 8.123721029730607122251098581030, 9.281555630135559416055333377262

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