Properties

Degree 1
Conductor $ 5^{2} $
Sign $0.968 + 0.248i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.809 + 0.587i)2-s + (−0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (0.309 − 0.951i)6-s − 7-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + (−0.809 − 0.587i)11-s + (0.809 − 0.587i)12-s + (0.809 − 0.587i)13-s + (−0.809 − 0.587i)14-s + (−0.809 + 0.587i)16-s + (−0.309 + 0.951i)17-s − 18-s + (0.309 − 0.951i)19-s + ⋯
L(s,χ)  = 1  + (0.809 + 0.587i)2-s + (−0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (0.309 − 0.951i)6-s − 7-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + (−0.809 − 0.587i)11-s + (0.809 − 0.587i)12-s + (0.809 − 0.587i)13-s + (−0.809 − 0.587i)14-s + (−0.809 + 0.587i)16-s + (−0.309 + 0.951i)17-s − 18-s + (0.309 − 0.951i)19-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.968 + 0.248i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 25 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.968 + 0.248i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(25\)    =    \(5^{2}\)
\( \varepsilon \)  =  $0.968 + 0.248i$
motivic weight  =  \(0\)
character  :  $\chi_{25} (4, \cdot )$
Sato-Tate  :  $\mu(10)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 25,\ (0:\ ),\ 0.968 + 0.248i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8746047634 + 0.1104882761i$
$L(\frac12,\chi)$  $\approx$  $0.8746047634 + 0.1104882761i$
$L(\chi,1)$  $\approx$  1.137110429 + 0.1227687919i
$L(1,\chi)$  $\approx$  1.137110429 + 0.1227687919i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−38.566281692971408793603972926856, −37.701827944379050308506558826541, −35.95065728837175317528230347978, −34.02178254415774019504554443668, −33.08680663122373125628669968924, −31.993422626565167957055373117046, −30.94902972217478728491446142613, −29.03141770700617946926343968269, −28.537804100970896404196821387387, −26.88689509289986965592410580026, −25.350915097664899028332910338267, −23.28624379966554812817764368891, −22.62593152657361199528143198695, −21.21288945616132045470134485323, −20.26784925989538997233437077417, −18.58274875785159250852461552240, −16.32069126609366156415811923032, −15.34497249886866747459541622087, −13.67535680257078377851095147170, −12.11091820547122992104187370874, −10.61201408560344711140493055722, −9.46508246406389451808855605233, −6.33072922106933373242898651201, −4.720844063769540651069167836267, −3.12335403531942608139244869971, 3.06516421169049963067569008286, 5.58061874668819666636236559749, 6.781405800920544365901057691297, 8.33490915515581935912059211202, 11.15123966521173621738261783094, 12.891511748596364438957922994, 13.44239159278436398126373315317, 15.45222729004562975582254322076, 16.77292078493330666291596245636, 18.24473803468583995215885283700, 19.86038415952044778886940457548, 21.736099922508683650078333968298, 23.03346078159664038100689143292, 23.894149400292464899067347399555, 25.255789904816721094420869637168, 26.23136205482279920852304648021, 28.58024322793023298395798562663, 29.70781264532335625179810739401, 30.840750783250949686215772805529, 32.06619796099978803452237714237, 33.35389369815977330575069087417, 34.91325396356181924805760772548, 35.31476693130028434345842290738, 36.96413157370548067780585986765, 38.979590011150217333522778791294

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