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Properties

Label 1-920-920.683-r0-0-0
Degree $1$
Conductor $920$
Sign $0.276 + 0.961i$
Analytic cond. $4.27246$
Root an. cond. $4.27246$
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  + (0.909 + 0.415i)3-s + (0.540 + 0.841i)7-s + (0.654 + 0.755i)9-s + (−0.142 + 0.989i)11-s + (0.540 − 0.841i)13-s + (−0.281 + 0.959i)17-s + (0.959 − 0.281i)19-s + (0.142 + 0.989i)21-s + (0.281 + 0.959i)27-s + (−0.959 − 0.281i)29-s + (−0.415 − 0.909i)31-s + (−0.540 + 0.841i)33-s + (0.755 − 0.654i)37-s + (0.841 − 0.540i)39-s + (−0.654 + 0.755i)41-s + ⋯
L(s)  = 1  + (0.909 + 0.415i)3-s + (0.540 + 0.841i)7-s + (0.654 + 0.755i)9-s + (−0.142 + 0.989i)11-s + (0.540 − 0.841i)13-s + (−0.281 + 0.959i)17-s + (0.959 − 0.281i)19-s + (0.142 + 0.989i)21-s + (0.281 + 0.959i)27-s + (−0.959 − 0.281i)29-s + (−0.415 − 0.909i)31-s + (−0.540 + 0.841i)33-s + (0.755 − 0.654i)37-s + (0.841 − 0.540i)39-s + (−0.654 + 0.755i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.276 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.276 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.276 + 0.961i$
Analytic conductor: \(4.27246\)
Root analytic conductor: \(4.27246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (0:\ ),\ 0.276 + 0.961i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.793909025 + 1.351106112i\)
\(L(\frac12)\) \(\approx\) \(1.793909025 + 1.351106112i\)
\(L(1)\) \(\approx\) \(1.466677188 + 0.5078856391i\)
\(L(1)\) \(\approx\) \(1.466677188 + 0.5078856391i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.909 + 0.415i)T \)
7 \( 1 + (0.540 + 0.841i)T \)
11 \( 1 + (-0.142 + 0.989i)T \)
13 \( 1 + (0.540 - 0.841i)T \)
17 \( 1 + (-0.281 + 0.959i)T \)
19 \( 1 + (0.959 - 0.281i)T \)
29 \( 1 + (-0.959 - 0.281i)T \)
31 \( 1 + (-0.415 - 0.909i)T \)
37 \( 1 + (0.755 - 0.654i)T \)
41 \( 1 + (-0.654 + 0.755i)T \)
43 \( 1 + (0.909 + 0.415i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.540 - 0.841i)T \)
59 \( 1 + (-0.841 - 0.540i)T \)
61 \( 1 + (-0.415 - 0.909i)T \)
67 \( 1 + (-0.989 + 0.142i)T \)
71 \( 1 + (0.142 + 0.989i)T \)
73 \( 1 + (-0.281 - 0.959i)T \)
79 \( 1 + (0.841 + 0.540i)T \)
83 \( 1 + (0.755 - 0.654i)T \)
89 \( 1 + (-0.415 + 0.909i)T \)
97 \( 1 + (0.755 + 0.654i)T \)
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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

−21.49465932217547204510342916645, −20.747297579706414826017590541514, −20.24359727119216193555202456353, −19.41079026043855262084582921696, −18.45781076203359160523631290482, −18.150652523197135845815564130619, −16.84647920655242635729635086656, −16.19815392205733369513713131537, −15.270339488236272059380360202737, −14.130457846592803937141664228317, −13.89516820235116762930201980016, −13.22254407635936027796600919934, −11.999132052957877301378260364590, −11.2531341734570839953005872804, −10.33943158125301498721902928462, −9.21253216702008984261700309009, −8.67965739070328443173231496825, −7.60478726113721429837362626492, −7.13991109156406040572789425045, −6.03536724049604310752666034690, −4.813947742202994180329761003741, −3.7855391059529413584482566297, −3.08624045715004311112256698215, −1.8246684742316543208474005819, −0.937511104183572619123201558314, 1.56477551151466920285606448301, 2.37374260391729114209951926369, 3.349706835713283224352313001134, 4.35553758257124977454636176483, 5.21450103866311048427731440426, 6.17567215444127178383229302685, 7.67160974213303513335366514810, 7.94975501401511297276331143795, 9.09501817374698840386786467996, 9.603053210136638379053064393946, 10.65323468295295886343270520243, 11.42455607455268977364452228288, 12.66483072050728952782434305221, 13.122679746161307701950209427022, 14.25403574288948477865434307545, 15.044050343161128837195714494656, 15.37508201442439552868806082970, 16.25153573304112241768800481043, 17.467910328265810885853729357911, 18.15071079431482603473132455094, 18.89863913612703235054008110834, 19.88556972430603544756478126285, 20.49218787047570826422640151641, 21.08800967863263769932136818561, 22.027750795354639500408044038031

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