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Properties

Label 1-632-632.397-r0-0-0
Degree $1$
Conductor $632$
Sign $0.773 - 0.633i$
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  + (−0.948 − 0.316i)3-s + (−0.428 − 0.903i)5-s + (−0.200 − 0.979i)7-s + (0.799 + 0.600i)9-s + (0.996 − 0.0804i)11-s + (0.0402 + 0.999i)13-s + (0.120 + 0.992i)15-s + (0.885 + 0.464i)17-s + (0.632 + 0.774i)19-s + (−0.120 + 0.992i)21-s + (−0.5 + 0.866i)23-s + (−0.632 + 0.774i)25-s + (−0.568 − 0.822i)27-s + (0.919 + 0.391i)29-s + (0.692 − 0.721i)31-s + ⋯
L(s)  = 1  + (−0.948 − 0.316i)3-s + (−0.428 − 0.903i)5-s + (−0.200 − 0.979i)7-s + (0.799 + 0.600i)9-s + (0.996 − 0.0804i)11-s + (0.0402 + 0.999i)13-s + (0.120 + 0.992i)15-s + (0.885 + 0.464i)17-s + (0.632 + 0.774i)19-s + (−0.120 + 0.992i)21-s + (−0.5 + 0.866i)23-s + (−0.632 + 0.774i)25-s + (−0.568 − 0.822i)27-s + (0.919 + 0.391i)29-s + (0.692 − 0.721i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 632 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 632 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(632\)    =    \(2^{3} \cdot 79\)
Sign: $0.773 - 0.633i$
Analytic conductor: \(2.93499\)
Root analytic conductor: \(2.93499\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{632} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 632,\ (0:\ ),\ 0.773 - 0.633i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9482075482 - 0.3384887382i\)
\(L(\frac12)\) \(\approx\) \(0.9482075482 - 0.3384887382i\)
\(L(1)\) \(\approx\) \(0.8132202361 - 0.1987980269i\)
\(L(1)\) \(\approx\) \(0.8132202361 - 0.1987980269i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
79 \( 1 \)
good3 \( 1 + (-0.948 - 0.316i)T \)
5 \( 1 + (-0.428 - 0.903i)T \)
7 \( 1 + (-0.200 - 0.979i)T \)
11 \( 1 + (0.996 - 0.0804i)T \)
13 \( 1 + (0.0402 + 0.999i)T \)
17 \( 1 + (0.885 + 0.464i)T \)
19 \( 1 + (0.632 + 0.774i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.919 + 0.391i)T \)
31 \( 1 + (0.692 - 0.721i)T \)
37 \( 1 + (-0.987 + 0.160i)T \)
41 \( 1 + (0.568 - 0.822i)T \)
43 \( 1 + (0.996 + 0.0804i)T \)
47 \( 1 + (0.987 + 0.160i)T \)
53 \( 1 + (-0.948 + 0.316i)T \)
59 \( 1 + (0.845 + 0.534i)T \)
61 \( 1 + (0.354 - 0.935i)T \)
67 \( 1 + (0.970 - 0.239i)T \)
71 \( 1 + (-0.748 - 0.663i)T \)
73 \( 1 + (-0.0402 + 0.999i)T \)
83 \( 1 + (0.845 - 0.534i)T \)
89 \( 1 + (-0.748 + 0.663i)T \)
97 \( 1 + (-0.354 + 0.935i)T \)
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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

−22.62908499633753870932230774069, −22.48410094916446457689098947943, −21.70290936575259421814621732697, −20.719461470611378809685460007639, −19.55427191680638964172577455078, −18.83708102365295042761157685170, −17.938580933071519274723791706646, −17.497099663432792473675240976270, −16.1426590301220456216395885408, −15.70863454155341343449810361845, −14.85251935841065088468542426621, −14.026767253035777748826043078014, −12.524546786299195500228053386364, −12.004040889418687110670250088041, −11.30132507558259812356854980425, −10.3220844110383623845094538506, −9.63569812073746657967633968742, −8.47137778280629380569426882291, −7.271130714444087707189945750048, −6.43560001236465744156282615301, −5.69233600219898788046679163660, −4.6668193886327052324609246024, −3.4954718520739350493480970414, −2.61779073791890082337127344144, −0.85980302873919316208076248041, 0.93447628208731244565725121273, 1.58606590345676915439313260245, 3.757705233668281702825577725258, 4.26131400172922591426532553041, 5.40037109813103771605895851866, 6.32781965022844551695767561718, 7.27011666226581280506629615897, 8.03226399262619736325821699092, 9.31164710904169003626245183302, 10.111619570566162517807700521152, 11.190141342001822393592684809136, 12.02221651742399888412998266984, 12.439232368003445302639550644964, 13.66020866351359848216478284149, 14.2113763553967798878900262139, 15.80601255331196175341581094468, 16.34431040076901951973234623967, 17.09636400105031503245018281605, 17.5061637649831043612150679985, 19.01268639302204046306440576632, 19.332930413358908012259231495341, 20.41951761738738249848134968854, 21.20470519357633169705517606974, 22.18948870094995460836779268798, 23.029938144051487807414397460962

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