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Properties

Label 2-644-7.4-c1-0-6
Degree $2$
Conductor $644$
Sign $0.640 - 0.767i$
Analytic cond. $5.14236$
Root an. cond. $2.26767$
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  + (−0.331 + 0.573i)3-s + (1.49 + 2.59i)5-s + (2.42 − 1.06i)7-s + (1.28 + 2.21i)9-s + (2.82 − 4.89i)11-s − 1.87·13-s − 1.98·15-s + (0.0960 − 0.166i)17-s + (0.986 + 1.70i)19-s + (−0.189 + 1.74i)21-s + (0.5 + 0.866i)23-s + (−1.97 + 3.42i)25-s − 3.68·27-s − 0.907·29-s + (−1.91 + 3.31i)31-s + ⋯
L(s)  = 1  + (−0.191 + 0.331i)3-s + (0.669 + 1.15i)5-s + (0.915 − 0.403i)7-s + (0.426 + 0.739i)9-s + (0.852 − 1.47i)11-s − 0.521·13-s − 0.511·15-s + (0.0232 − 0.0403i)17-s + (0.226 + 0.391i)19-s + (−0.0414 + 0.380i)21-s + (0.104 + 0.180i)23-s + (−0.395 + 0.684i)25-s − 0.708·27-s − 0.168·29-s + (−0.343 + 0.594i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 - 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.640 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $0.640 - 0.767i$
Analytic conductor: \(5.14236\)
Root analytic conductor: \(2.26767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{644} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 644,\ (\ :1/2),\ 0.640 - 0.767i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63962 + 0.767104i\)
\(L(\frac12)\) \(\approx\) \(1.63962 + 0.767104i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-2.42 + 1.06i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (0.331 - 0.573i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.49 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.82 + 4.89i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.87T + 13T^{2} \)
17 \( 1 + (-0.0960 + 0.166i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.986 - 1.70i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 + 0.907T + 29T^{2} \)
31 \( 1 + (1.91 - 3.31i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.941 - 1.63i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.12T + 41T^{2} \)
43 \( 1 - 9.84T + 43T^{2} \)
47 \( 1 + (-2.08 - 3.61i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.178 - 0.310i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.10 - 8.83i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.46 + 9.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.52 + 9.57i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 15.6T + 71T^{2} \)
73 \( 1 + (3.22 - 5.58i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.23 - 3.86i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.10T + 83T^{2} \)
89 \( 1 + (0.163 + 0.282i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.09T + 97T^{2} \)
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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

−10.81561957393005272160863170127, −10.04499244300017676326043222294, −9.059971845138371012419149416278, −7.949709259387656131149732029296, −7.14933616796633004636988891752, −6.15279295410392630830924727259, −5.28764119651749127417160112593, −4.12522162541071698650075725691, −2.96145434205543956216973939723, −1.59009471461584419158468951490, 1.23449456237805614134743020980, 2.12127766552681894390218266731, 4.18130475220740173534537731929, 4.89923876619397457398217202844, 5.82433549847124267409229610658, 6.93626163158519626116527386223, 7.73668319693025820300483642457, 9.088661267660301739924343270629, 9.263071305459250938087005032638, 10.27517547108042065951279325472

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