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Properties

Label 1-1625-1625.246-r0-0-0
Degree $1$
Conductor $1625$
Sign $0.762 - 0.647i$
Analytic cond. $7.54646$
Root an. cond. $7.54646$
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  + (0.992 + 0.125i)2-s + (−0.637 − 0.770i)3-s + (0.968 + 0.248i)4-s + (−0.535 − 0.844i)6-s + (−0.309 − 0.951i)7-s + (0.929 + 0.368i)8-s + (−0.187 + 0.982i)9-s + (0.992 + 0.125i)11-s + (−0.425 − 0.904i)12-s + (−0.187 − 0.982i)14-s + (0.876 + 0.481i)16-s + (0.968 − 0.248i)17-s + (−0.309 + 0.951i)18-s + (0.637 − 0.770i)19-s + (−0.535 + 0.844i)21-s + (0.968 + 0.248i)22-s + ⋯
L(s)  = 1  + (0.992 + 0.125i)2-s + (−0.637 − 0.770i)3-s + (0.968 + 0.248i)4-s + (−0.535 − 0.844i)6-s + (−0.309 − 0.951i)7-s + (0.929 + 0.368i)8-s + (−0.187 + 0.982i)9-s + (0.992 + 0.125i)11-s + (−0.425 − 0.904i)12-s + (−0.187 − 0.982i)14-s + (0.876 + 0.481i)16-s + (0.968 − 0.248i)17-s + (−0.309 + 0.951i)18-s + (0.637 − 0.770i)19-s + (−0.535 + 0.844i)21-s + (0.968 + 0.248i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1625 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.762 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1625 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.762 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1625\)    =    \(5^{3} \cdot 13\)
Sign: $0.762 - 0.647i$
Analytic conductor: \(7.54646\)
Root analytic conductor: \(7.54646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1625} (246, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1625,\ (0:\ ),\ 0.762 - 0.647i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.637132195 - 0.9681859678i\)
\(L(\frac12)\) \(\approx\) \(2.637132195 - 0.9681859678i\)
\(L(1)\) \(\approx\) \(1.736582318 - 0.3658530771i\)
\(L(1)\) \(\approx\) \(1.736582318 - 0.3658530771i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.992 + 0.125i)T \)
3 \( 1 + (-0.637 - 0.770i)T \)
7 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (0.992 + 0.125i)T \)
17 \( 1 + (0.968 - 0.248i)T \)
19 \( 1 + (0.637 - 0.770i)T \)
23 \( 1 + (0.728 + 0.684i)T \)
29 \( 1 + (0.0627 + 0.998i)T \)
31 \( 1 + (-0.968 + 0.248i)T \)
37 \( 1 + (-0.876 - 0.481i)T \)
41 \( 1 + (-0.728 + 0.684i)T \)
43 \( 1 + (-0.809 + 0.587i)T \)
47 \( 1 + (0.929 - 0.368i)T \)
53 \( 1 + (0.535 - 0.844i)T \)
59 \( 1 + (0.425 + 0.904i)T \)
61 \( 1 + (0.728 + 0.684i)T \)
67 \( 1 + (-0.0627 + 0.998i)T \)
71 \( 1 + (0.929 - 0.368i)T \)
73 \( 1 + (0.425 - 0.904i)T \)
79 \( 1 + (-0.637 - 0.770i)T \)
83 \( 1 + (0.637 - 0.770i)T \)
89 \( 1 + (0.425 - 0.904i)T \)
97 \( 1 + (-0.0627 - 0.998i)T \)
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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

−20.765292792895041001825519987522, −20.09486931912435539449791573881, −18.99767346851317539087309400934, −18.53007280478317033039724688958, −17.04509068757842456482513123977, −16.80757674829113919528395003230, −15.84694444019140972415601169345, −15.2950260038289141932344017264, −14.58766523201920427580022776674, −13.96199918430604796125256040910, −12.730944610161723658880725301617, −12.10699169912581528521118784386, −11.71860584590553348015756340684, −10.81040043218879864178249275960, −9.98175132585001190497464630532, −9.310935262677330849375137156206, −8.276009769311622179717285709068, −6.96838129281003115901108250774, −6.26011677671167501404066918006, −5.53766927087378822246838426174, −5.02721896657507210837366662259, −3.7853068781858471828684145259, −3.479014668659751554106539222531, −2.28115583800112869171155699582, −1.08607363740333626538800502, 1.00067117635609736750345065896, 1.70229166242799684056777575622, 3.06756656576168300826366899098, 3.7264099553691375618887607299, 4.86601659939005244312987170498, 5.4455871414320057824931341599, 6.434208909506110338649843013294, 7.25820066823260564496785723017, 7.30565158924660729848442360920, 8.70418052843411531976000197978, 9.94300099260536165378772312566, 10.781684324948209508290620806059, 11.538342303664174617782598916337, 12.06867379695051994797507896291, 12.96251331636484522225897307464, 13.48552389790202442886327556113, 14.2022426746968312796811664353, 14.86675018740360477982072204100, 16.10808724839842626804858205102, 16.557818539844669848905536378117, 17.20951483482588121185532300858, 17.94596181156923787220017773208, 19.07053516928478527997194251586, 19.77285905590725822695420295262, 20.205232561963290290896809784847

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