Properties

Label 1-920-920.659-r0-0-0
Degree $1$
Conductor $920$
Sign $-0.552 - 0.833i$
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  + (0.654 − 0.755i)3-s + (−0.415 − 0.909i)7-s + (−0.142 − 0.989i)9-s + (0.959 − 0.281i)11-s + (0.415 − 0.909i)13-s + (0.841 − 0.540i)17-s + (−0.841 − 0.540i)19-s + (−0.959 − 0.281i)21-s + (−0.841 − 0.540i)27-s + (−0.841 + 0.540i)29-s + (0.654 + 0.755i)31-s + (0.415 − 0.909i)33-s + (0.142 + 0.989i)37-s + (−0.415 − 0.909i)39-s + (−0.142 + 0.989i)41-s + ⋯
L(s)  = 1  + (0.654 − 0.755i)3-s + (−0.415 − 0.909i)7-s + (−0.142 − 0.989i)9-s + (0.959 − 0.281i)11-s + (0.415 − 0.909i)13-s + (0.841 − 0.540i)17-s + (−0.841 − 0.540i)19-s + (−0.959 − 0.281i)21-s + (−0.841 − 0.540i)27-s + (−0.841 + 0.540i)29-s + (0.654 + 0.755i)31-s + (0.415 − 0.909i)33-s + (0.142 + 0.989i)37-s + (−0.415 − 0.909i)39-s + (−0.142 + 0.989i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.552 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.552 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $-0.552 - 0.833i$
Analytic conductor: \(4.27246\)
Root analytic conductor: \(4.27246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (659, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (0:\ ),\ -0.552 - 0.833i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8304828674 - 1.546009733i\)
\(L(\frac12)\) \(\approx\) \(0.8304828674 - 1.546009733i\)
\(L(1)\) \(\approx\) \(1.123430644 - 0.6562973811i\)
\(L(1)\) \(\approx\) \(1.123430644 - 0.6562973811i\)

Euler product

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

−22.10194851007909019499457656144, −21.22858958617026219830057193543, −20.83121214851768427183904498535, −19.6730471752749194048005854993, −19.100160469340180176807741079160, −18.53298502659045635667338355534, −17.03503450446859729789568486420, −16.669476738193855098832537432640, −15.63132138262275999492409434754, −15.003428002918923731414525832670, −14.34185382957704516470829497923, −13.50081173786245389364941496383, −12.41900451839308382067558539073, −11.7244187863214381158058034211, −10.67464808717747481291923034116, −9.77683922222497527067941071131, −9.08138224780955487984916013311, −8.52468642444602166906599865161, −7.43401494213369001901867217047, −6.24159977433605699955906411247, −5.54128713977052207467068859748, −4.16383178851885119363535112670, −3.777801059551364194036617301187, −2.510254204978144150384068304139, −1.697336496903967719325983164726, 0.72683753258464995551919043143, 1.58527079584711973963637760892, 3.070908729996646041493133862623, 3.49911632574619300367154366320, 4.717904928001687219968466521848, 6.133110291515317641895059757281, 6.72188637264118918148709298918, 7.6169033453402379562856083972, 8.37904679259957583894161294913, 9.28838207580115086932288543313, 10.12235640040857289606637713142, 11.13882837910128440300260898990, 12.07337735857845933258664803812, 12.981978228233433134602195027762, 13.50776370142158554154923642073, 14.350289360341810286328741646531, 14.99848166105142066606424374441, 16.10259616662667312642235407762, 16.98933704669058698974956322059, 17.64853302671705806751603792317, 18.62320110017004780741425334470, 19.29577874071587781028669222498, 20.05747962241813496157170589780, 20.492746692935660975000383008412, 21.52938713206353172060901537214

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