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Properties

Label 1-632-632.413-r0-0-0
Degree $1$
Conductor $632$
Sign $0.783 + 0.621i$
Analytic cond. $2.93499$
Root an. cond. $2.93499$
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  + (−0.885 − 0.464i)3-s + (−0.120 + 0.992i)5-s + (0.885 + 0.464i)7-s + (0.568 + 0.822i)9-s + (−0.120 − 0.992i)11-s + (0.748 + 0.663i)13-s + (0.568 − 0.822i)15-s + (−0.748 − 0.663i)17-s + (0.970 − 0.239i)19-s + (−0.568 − 0.822i)21-s + 23-s + (−0.970 − 0.239i)25-s + (−0.120 − 0.992i)27-s + (−0.568 + 0.822i)29-s + (−0.354 + 0.935i)31-s + ⋯
L(s)  = 1  + (−0.885 − 0.464i)3-s + (−0.120 + 0.992i)5-s + (0.885 + 0.464i)7-s + (0.568 + 0.822i)9-s + (−0.120 − 0.992i)11-s + (0.748 + 0.663i)13-s + (0.568 − 0.822i)15-s + (−0.748 − 0.663i)17-s + (0.970 − 0.239i)19-s + (−0.568 − 0.822i)21-s + 23-s + (−0.970 − 0.239i)25-s + (−0.120 − 0.992i)27-s + (−0.568 + 0.822i)29-s + (−0.354 + 0.935i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 632 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 632 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(632\)    =    \(2^{3} \cdot 79\)
Sign: $0.783 + 0.621i$
Analytic conductor: \(2.93499\)
Root analytic conductor: \(2.93499\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{632} (413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 632,\ (0:\ ),\ 0.783 + 0.621i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.074165817 + 0.3742520894i\)
\(L(\frac12)\) \(\approx\) \(1.074165817 + 0.3742520894i\)
\(L(1)\) \(\approx\) \(0.9192747550 + 0.1069973963i\)
\(L(1)\) \(\approx\) \(0.9192747550 + 0.1069973963i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
79 \( 1 \)
good3 \( 1 + (-0.885 - 0.464i)T \)
5 \( 1 + (-0.120 + 0.992i)T \)
7 \( 1 + (0.885 + 0.464i)T \)
11 \( 1 + (-0.120 - 0.992i)T \)
13 \( 1 + (0.748 + 0.663i)T \)
17 \( 1 + (-0.748 - 0.663i)T \)
19 \( 1 + (0.970 - 0.239i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.568 + 0.822i)T \)
31 \( 1 + (-0.354 + 0.935i)T \)
37 \( 1 + (0.970 - 0.239i)T \)
41 \( 1 + (0.120 - 0.992i)T \)
43 \( 1 + (-0.120 + 0.992i)T \)
47 \( 1 + (-0.970 - 0.239i)T \)
53 \( 1 + (-0.885 + 0.464i)T \)
59 \( 1 + (0.748 - 0.663i)T \)
61 \( 1 + (0.970 - 0.239i)T \)
67 \( 1 + (0.354 + 0.935i)T \)
71 \( 1 + (0.885 - 0.464i)T \)
73 \( 1 + (-0.748 + 0.663i)T \)
83 \( 1 + (0.748 + 0.663i)T \)
89 \( 1 + (0.885 + 0.464i)T \)
97 \( 1 + (-0.970 + 0.239i)T \)
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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

−22.95771634439434952038992430838, −22.11615861731240892516561411936, −20.934586887914457441297519180930, −20.66635223847968472045408860928, −19.89754446933616964651163482691, −18.427315779796238943642713327, −17.65715892137843545046897027685, −17.14580511640678650643899663117, −16.32700717489476688078625716142, −15.40538695867262440068448097793, −14.83397237389512036055216406518, −13.29060760204913922707264560828, −12.83470884444234035065745700422, −11.664739034215431927716437338510, −11.168278272721064264298273066531, −10.12185450767576947189974704977, −9.32302200719286304774001830469, −8.20666796946561454147902508810, −7.3828119689963299293115932215, −6.10943009243779783015025605660, −5.173887018176125925041706015733, −4.536857845021653297262076584613, −3.71436575985986712290451948604, −1.786487510106044172766847500764, −0.80235811583881919165096457876, 1.1231927488604074991867230483, 2.29937591059614894751566938350, 3.444007602422068780210375737009, 4.82488117109101493241935374603, 5.62848283499919691306492909531, 6.598529479832503841439163788221, 7.28331330043050552186731680833, 8.31503265018311730575703734755, 9.35718797202792346097296063861, 10.82523110674296239841509945734, 11.21182882928364024205797848363, 11.67466408851385725339199239916, 12.97259965857498761346653324187, 13.859114781600348787594704386969, 14.54936673625828522921521772088, 15.76720794183430674533929826688, 16.28899069934111941349785242755, 17.511998745428669196466314814683, 18.21029515011376352304206594713, 18.60675723434785135183249904090, 19.441942336253712416372751223863, 20.747207219256346073818279795505, 21.72910272725242551330217942323, 22.09790537187525441874854049573, 23.11364629412617712596657980940

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