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Properties

Label 1-5e4-625.131-r0-0-0
Degree $1$
Conductor $625$
Sign $-0.993 - 0.115i$
Analytic cond. $2.90248$
Root an. cond. $2.90248$
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  + (−0.745 − 0.666i)2-s + (0.850 − 0.525i)3-s + (0.112 + 0.993i)4-s + (−0.984 − 0.175i)6-s + (0.0627 − 0.998i)7-s + (0.577 − 0.816i)8-s + (0.448 − 0.893i)9-s + (0.212 + 0.977i)11-s + (0.617 + 0.786i)12-s + (−0.711 − 0.702i)13-s + (−0.711 + 0.702i)14-s + (−0.974 + 0.224i)16-s + (−0.910 − 0.414i)17-s + (−0.929 + 0.368i)18-s + (−0.997 + 0.0753i)19-s + ⋯
L(s)  = 1  + (−0.745 − 0.666i)2-s + (0.850 − 0.525i)3-s + (0.112 + 0.993i)4-s + (−0.984 − 0.175i)6-s + (0.0627 − 0.998i)7-s + (0.577 − 0.816i)8-s + (0.448 − 0.893i)9-s + (0.212 + 0.977i)11-s + (0.617 + 0.786i)12-s + (−0.711 − 0.702i)13-s + (−0.711 + 0.702i)14-s + (−0.974 + 0.224i)16-s + (−0.910 − 0.414i)17-s + (−0.929 + 0.368i)18-s + (−0.997 + 0.0753i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $-0.993 - 0.115i$
Analytic conductor: \(2.90248\)
Root analytic conductor: \(2.90248\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 625,\ (0:\ ),\ -0.993 - 0.115i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05005439766 - 0.8649488817i\)
\(L(\frac12)\) \(\approx\) \(0.05005439766 - 0.8649488817i\)
\(L(1)\) \(\approx\) \(0.6513950822 - 0.5305958315i\)
\(L(1)\) \(\approx\) \(0.6513950822 - 0.5305958315i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + (-0.745 - 0.666i)T \)
3 \( 1 + (0.850 - 0.525i)T \)
7 \( 1 + (0.0627 - 0.998i)T \)
11 \( 1 + (0.212 + 0.977i)T \)
13 \( 1 + (-0.711 - 0.702i)T \)
17 \( 1 + (-0.910 - 0.414i)T \)
19 \( 1 + (-0.997 + 0.0753i)T \)
23 \( 1 + (-0.999 + 0.0251i)T \)
29 \( 1 + (0.260 - 0.965i)T \)
31 \( 1 + (-0.910 - 0.414i)T \)
37 \( 1 + (-0.514 - 0.857i)T \)
41 \( 1 + (0.793 + 0.607i)T \)
43 \( 1 + (0.728 + 0.684i)T \)
47 \( 1 + (0.0125 - 0.999i)T \)
53 \( 1 + (0.899 + 0.437i)T \)
59 \( 1 + (-0.962 - 0.272i)T \)
61 \( 1 + (0.793 - 0.607i)T \)
67 \( 1 + (0.998 + 0.0502i)T \)
71 \( 1 + (-0.947 - 0.320i)T \)
73 \( 1 + (-0.0376 + 0.999i)T \)
79 \( 1 + (0.850 - 0.525i)T \)
83 \( 1 + (-0.379 - 0.925i)T \)
89 \( 1 + (0.617 - 0.786i)T \)
97 \( 1 + (-0.837 - 0.546i)T \)
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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

−23.98329464116214298364847000501, −22.24257386443141182795204204202, −21.816249747487164907020073404172, −20.85899069358929559217497480202, −19.6999740647137705735827973726, −19.299193170508323301016089384053, −18.546745252598265318817131325269, −17.55922221548688133491099346017, −16.52126387626720117352629384710, −15.92636471143814115668449540185, −15.10383413463675088584066553825, −14.44492162993437618494598760355, −13.73000830524797660681774006338, −12.444792269326902809495010563488, −11.18193723406973909964786663699, −10.44345969436986440577654745006, −9.32130772160020248889904212599, −8.80284713610871741463595187751, −8.23175864881505317428825146986, −7.04991655411057713644508018423, −6.075628498009154840457433440893, −5.07103755981023912513855389230, −4.02498066798846062968343232843, −2.55328269647457230289665703861, −1.79800160267084607313348997166, 0.462188049495933702413715819233, 1.871972221852932350495114191542, 2.493285935007057039223461285871, 3.81679012892589855234155151784, 4.445708738221544438225467504058, 6.48142226162866963340103003274, 7.397521411837733921808904430024, 7.836038209097998104220419999730, 8.92840027898035039913149943272, 9.78001077766700836522418863526, 10.42627611533744347865306935692, 11.55386540399665741498066875066, 12.59455118370335739605063272372, 13.06772697123884749964352214956, 14.06702163738988311832783083891, 14.99672978696948666962574219786, 15.99255601748466788732154121873, 17.26098960183861005946645001671, 17.64259561140925728793400094636, 18.49979208600169399574843500231, 19.67301591473098943904991350971, 19.88127310461216038930079492260, 20.56059829724574379379666790345, 21.43767004504007862440820515106, 22.53659070969007434048339640773

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