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Properties

Label 2-644-644.111-c0-0-0
Degree $2$
Conductor $644$
Sign $0.268 - 0.963i$
Analytic cond. $0.321397$
Root an. cond. $0.566919$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.959 − 0.281i)2-s + (0.841 + 0.540i)4-s + (0.142 + 0.989i)7-s + (−0.654 − 0.755i)8-s + (−0.841 + 0.540i)9-s + (−0.797 + 1.74i)11-s + (0.142 − 0.989i)14-s + (0.415 + 0.909i)16-s + (0.959 − 0.281i)18-s + (1.25 − 1.45i)22-s + (−0.841 − 0.540i)23-s + (−0.415 − 0.909i)25-s + (−0.415 + 0.909i)28-s + (0.544 + 0.627i)29-s + (−0.142 − 0.989i)32-s + ⋯
L(s)  = 1  + (−0.959 − 0.281i)2-s + (0.841 + 0.540i)4-s + (0.142 + 0.989i)7-s + (−0.654 − 0.755i)8-s + (−0.841 + 0.540i)9-s + (−0.797 + 1.74i)11-s + (0.142 − 0.989i)14-s + (0.415 + 0.909i)16-s + (0.959 − 0.281i)18-s + (1.25 − 1.45i)22-s + (−0.841 − 0.540i)23-s + (−0.415 − 0.909i)25-s + (−0.415 + 0.909i)28-s + (0.544 + 0.627i)29-s + (−0.142 − 0.989i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.268 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.268 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $0.268 - 0.963i$
Analytic conductor: \(0.321397\)
Root analytic conductor: \(0.566919\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{644} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 644,\ (\ :0),\ 0.268 - 0.963i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5024921180\)
\(L(\frac12)\) \(\approx\) \(0.5024921180\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.959 + 0.281i)T \)
7 \( 1 + (-0.142 - 0.989i)T \)
23 \( 1 + (0.841 + 0.540i)T \)
good3 \( 1 + (0.841 - 0.540i)T^{2} \)
5 \( 1 + (0.415 + 0.909i)T^{2} \)
11 \( 1 + (0.797 - 1.74i)T + (-0.654 - 0.755i)T^{2} \)
13 \( 1 + (0.959 + 0.281i)T^{2} \)
17 \( 1 + (-0.142 + 0.989i)T^{2} \)
19 \( 1 + (0.142 + 0.989i)T^{2} \)
29 \( 1 + (-0.544 - 0.627i)T + (-0.142 + 0.989i)T^{2} \)
31 \( 1 + (0.841 + 0.540i)T^{2} \)
37 \( 1 + (-1.07 - 1.66i)T + (-0.415 + 0.909i)T^{2} \)
41 \( 1 + (-0.415 - 0.909i)T^{2} \)
43 \( 1 + (-1.61 + 0.474i)T + (0.841 - 0.540i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-1.80 + 0.258i)T + (0.959 - 0.281i)T^{2} \)
59 \( 1 + (-0.959 - 0.281i)T^{2} \)
61 \( 1 + (0.841 + 0.540i)T^{2} \)
67 \( 1 + (0.544 + 1.19i)T + (-0.654 + 0.755i)T^{2} \)
71 \( 1 + (-0.512 + 0.234i)T + (0.654 - 0.755i)T^{2} \)
73 \( 1 + (0.142 + 0.989i)T^{2} \)
79 \( 1 + (0.118 - 0.822i)T + (-0.959 - 0.281i)T^{2} \)
83 \( 1 + (-0.415 + 0.909i)T^{2} \)
89 \( 1 + (0.841 - 0.540i)T^{2} \)
97 \( 1 + (0.415 + 0.909i)T^{2} \)
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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

−10.72714059903530991283524940055, −10.08811937337569064995992359671, −9.263530382141837567605391338946, −8.298107078573295340292737953525, −7.81518291193146728287538834409, −6.67328547601951153342885687026, −5.63502510349941568582166537588, −4.49714041298747189608046562033, −2.70040520850210957600545557958, −2.12086698561808136966235554386, 0.74783838647248295121560878657, 2.66031742071056105082296293243, 3.81537901138590059114919552132, 5.62341239771800104637131448091, 6.04144310678366812185103965576, 7.34683591727067532755450186531, 8.006864160143949998736948140436, 8.787410606099120401184550000704, 9.652817952518020038221798570309, 10.65113387893414392445005855415

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