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Properties

Label 2-2646-63.25-c1-0-5
Degree $2$
Conductor $2646$
Sign $-0.980 - 0.196i$
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 + (−2.18 + 3.78i)5-s + 8-s + (−2.18 + 3.78i)10-s + (−0.686 − 1.18i)11-s + (1 + 1.73i)13-s + 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 + (−0.813 + 1.40i)23-s + (−7.05 − 12.2i)25-s + (1 + 1.73i)26-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (−0.977 + 1.69i)5-s + 0.353·8-s + (−0.691 + 1.19i)10-s + (−0.206 − 0.358i)11-s + (0.277 + 0.480i)13-s + 0.250·16-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 + (−0.169 + 0.293i)23-s + (−1.41 − 2.44i)25-s + (0.196 + 0.339i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.196i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.980 - 0.196i$
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: \((2,\ 2646,\ (\ :1/2),\ -0.980 - 0.196i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.336769787\)
\(L(\frac12)\) \(\approx\) \(1.336769787\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.18 - 3.78i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.686 + 1.18i)T + (-5.5 + 9.52i)T^{2} \)
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 \( 1 + (0.813 - 1.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.37 - 7.57i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.31 + 4.00i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.05 + 7.02i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (4.37 - 7.57i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 10.1T + 59T^{2} \)
61 \( 1 + 3.11T + 61T^{2} \)
67 \( 1 + 2.11T + 67T^{2} \)
71 \( 1 - 7.11T + 71T^{2} \)
73 \( 1 + (-6.05 + 10.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 5.11T + 79T^{2} \)
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

−9.219869122120921414987277976871, −8.157350719049029825767383212714, −7.47130118227948902858178935970, −6.96940624184912748308433865765, −6.17088288233884057113469709628, −5.45006040169215263728179701353, −4.15746318067926414287639071907, −3.58114607299868927787255372876, −2.97761988022879330292587371592, −1.82621732159872897239774498627, 0.33233821964166627654584319102, 1.51220566864135965388312693317, 2.88227252241806168873924286273, 3.92562984901351513306687206763, 4.58213437757829267315906975994, 5.14746321289865992209818907034, 5.92857800614131572124202000090, 7.07180178730420327683951830154, 7.84480058522525110544269405571, 8.298715462153050256728081165118

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