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Properties

Label 2-648-216.59-c1-0-20
Degree $2$
Conductor $648$
Sign $0.592 + 0.805i$
Analytic cond. $5.17430$
Root an. cond. $2.27471$
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  + (−0.955 − 1.04i)2-s + (−0.174 + 1.99i)4-s + (3.00 − 1.09i)5-s + (4.58 − 0.808i)7-s + (2.24 − 1.72i)8-s + (−4.01 − 2.09i)10-s + (0.303 − 0.833i)11-s + (−1.22 + 1.45i)13-s + (−5.22 − 4.00i)14-s + (−3.93 − 0.693i)16-s + (4.03 + 2.32i)17-s + (−0.171 − 0.296i)19-s + (1.65 + 6.18i)20-s + (−1.15 + 0.480i)22-s + (−1.00 + 5.68i)23-s + ⋯
L(s)  = 1  + (−0.675 − 0.737i)2-s + (−0.0870 + 0.996i)4-s + (1.34 − 0.489i)5-s + (1.73 − 0.305i)7-s + (0.793 − 0.608i)8-s + (−1.27 − 0.661i)10-s + (0.0914 − 0.251i)11-s + (−0.338 + 0.403i)13-s + (−1.39 − 1.07i)14-s + (−0.984 − 0.173i)16-s + (0.978 + 0.564i)17-s + (−0.0393 − 0.0680i)19-s + (0.370 + 1.38i)20-s + (−0.247 + 0.102i)22-s + (−0.208 + 1.18i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.592 + 0.805i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (611, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ 0.592 + 0.805i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42052 - 0.718558i\)
\(L(\frac12)\) \(\approx\) \(1.42052 - 0.718558i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.955 + 1.04i)T \)
3 \( 1 \)
good5 \( 1 + (-3.00 + 1.09i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (-4.58 + 0.808i)T + (6.57 - 2.39i)T^{2} \)
11 \( 1 + (-0.303 + 0.833i)T + (-8.42 - 7.07i)T^{2} \)
13 \( 1 + (1.22 - 1.45i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (-4.03 - 2.32i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.171 + 0.296i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.00 - 5.68i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (4.16 - 3.49i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (7.80 + 1.37i)T + (29.1 + 10.6i)T^{2} \)
37 \( 1 + (-2.31 - 1.33i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.0346 - 0.0413i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (2.85 + 1.04i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (1.90 + 10.7i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 - 1.27T + 53T^{2} \)
59 \( 1 + (2.43 + 6.68i)T + (-45.1 + 37.9i)T^{2} \)
61 \( 1 + (8.19 - 1.44i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (6.31 + 5.29i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (0.186 - 0.323i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.29 - 10.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.54 - 3.03i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (8.90 + 10.6i)T + (-14.4 + 81.7i)T^{2} \)
89 \( 1 + (-6.35 + 3.66i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.833 + 0.303i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.39161590464464308271464364187, −9.584768016848813858746996460269, −8.873892054112873992403975528268, −8.007387182633979319049496850633, −7.22942850991483215699082251392, −5.67793576972618341176190962650, −4.91673637471176008464821073572, −3.66553742968176598140839604422, −1.93321598843227703732646162419, −1.46246607791186092639626506176, 1.49136275046929101159953529763, 2.42216915554159802012627567701, 4.66005399952779338603070171243, 5.46497986394693840885769193848, 6.12330065206866706501133590738, 7.35673299304360993797213487314, 7.947107645360857484015406367751, 8.987522243105264711131891756247, 9.708275122598492953173945339411, 10.54060802541915103820422097942

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