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Properties

Label 1-664-664.411-r0-0-0
Degree $1$
Conductor $664$
Sign $-0.684 + 0.729i$
Analytic cond. $3.08360$
Root an. cond. $3.08360$
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  + (0.0383 − 0.999i)3-s + (−0.543 − 0.839i)5-s + (−0.338 − 0.941i)7-s + (−0.997 − 0.0765i)9-s + (−0.859 − 0.511i)11-s + (0.720 − 0.693i)13-s + (−0.859 + 0.511i)15-s + (−0.665 + 0.746i)17-s + (−0.606 − 0.795i)19-s + (−0.953 + 0.301i)21-s + (0.973 − 0.227i)23-s + (−0.409 + 0.912i)25-s + (−0.114 + 0.993i)27-s + (0.264 − 0.964i)29-s + (−0.896 + 0.443i)31-s + ⋯
L(s)  = 1  + (0.0383 − 0.999i)3-s + (−0.543 − 0.839i)5-s + (−0.338 − 0.941i)7-s + (−0.997 − 0.0765i)9-s + (−0.859 − 0.511i)11-s + (0.720 − 0.693i)13-s + (−0.859 + 0.511i)15-s + (−0.665 + 0.746i)17-s + (−0.606 − 0.795i)19-s + (−0.953 + 0.301i)21-s + (0.973 − 0.227i)23-s + (−0.409 + 0.912i)25-s + (−0.114 + 0.993i)27-s + (0.264 − 0.964i)29-s + (−0.896 + 0.443i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 664 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.684 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 664 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.684 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(664\)    =    \(2^{3} \cdot 83\)
Sign: $-0.684 + 0.729i$
Analytic conductor: \(3.08360\)
Root analytic conductor: \(3.08360\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{664} (411, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 664,\ (0:\ ),\ -0.684 + 0.729i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2486568473 - 0.5743459134i\)
\(L(\frac12)\) \(\approx\) \(-0.2486568473 - 0.5743459134i\)
\(L(1)\) \(\approx\) \(0.5558181634 - 0.5254162289i\)
\(L(1)\) \(\approx\) \(0.5558181634 - 0.5254162289i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
83 \( 1 \)
good3 \( 1 + (0.0383 - 0.999i)T \)
5 \( 1 + (-0.543 - 0.839i)T \)
7 \( 1 + (-0.338 - 0.941i)T \)
11 \( 1 + (-0.859 - 0.511i)T \)
13 \( 1 + (0.720 - 0.693i)T \)
17 \( 1 + (-0.665 + 0.746i)T \)
19 \( 1 + (-0.606 - 0.795i)T \)
23 \( 1 + (0.973 - 0.227i)T \)
29 \( 1 + (0.264 - 0.964i)T \)
31 \( 1 + (-0.896 + 0.443i)T \)
37 \( 1 + (0.997 - 0.0765i)T \)
41 \( 1 + (0.988 - 0.152i)T \)
43 \( 1 + (0.114 + 0.993i)T \)
47 \( 1 + (-0.927 - 0.373i)T \)
53 \( 1 + (-0.927 + 0.373i)T \)
59 \( 1 + (0.477 - 0.878i)T \)
61 \( 1 + (0.771 + 0.636i)T \)
67 \( 1 + (-0.817 - 0.575i)T \)
71 \( 1 + (0.338 - 0.941i)T \)
73 \( 1 + (0.409 + 0.912i)T \)
79 \( 1 + (0.953 + 0.301i)T \)
89 \( 1 + (-0.190 + 0.981i)T \)
97 \( 1 + (-0.190 - 0.981i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.07615140301429052789718666664, −22.499064385135650956515728745270, −21.68369122739622894774791415748, −21.004373513973808878190930449552, −20.16287630886568350231576481657, −19.12631219456003752239961153788, −18.49023630613047509019278035820, −17.69463228200469453483443926174, −16.229911616410822142745468605957, −15.99623934395282344429618370346, −14.98667682949491049002356066581, −14.62297455013273208024356659801, −13.37162196258556408481146855011, −12.30794844091658684190909059234, −11.267837454307626591372931258803, −10.83973012906402654689536641329, −9.75085482542739129369194162694, −9.00135002812101088924752068829, −8.09450373401307245259533237311, −6.92653732054990077765225121321, −5.97133505937390635687590339676, −4.9713164277087592656134323359, −3.98078940054322974540069983960, −3.032956016735826168341924631557, −2.23830292931104348678696441127, 0.32385056624594103653002090753, 1.218484477880147020958310175944, 2.65615216229997734963718146716, 3.7134848796622448563093991406, 4.81168270056196177014230047761, 5.93122214463034873302046544913, 6.8125326531863338913003936209, 7.87809547097422525342400732349, 8.309222800978063691489000494258, 9.31830571686037212415396011574, 10.877348329408953184300657386218, 11.1419666694433161018994702375, 12.62482240969095670674753916354, 13.07624960106744039419304202698, 13.49766781489934145241704313735, 14.81017778154709186782497453378, 15.779995221330922804575487033707, 16.59649949864879767735069354639, 17.39214262212604382738022827446, 18.12703809480627031526440191065, 19.299583641773654675238051676121, 19.63725472711919619191895465617, 20.50422881992757287581595913979, 21.23256148583521538021719535023, 22.62396361413239940615511249149

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