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Properties

Label 1-1625-1625.1152-r0-0-0
Degree $1$
Conductor $1625$
Sign $0.759 - 0.650i$
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  + (−0.968 + 0.248i)2-s + (−0.982 + 0.187i)3-s + (0.876 − 0.481i)4-s + (0.904 − 0.425i)6-s + (−0.809 − 0.587i)7-s + (−0.728 + 0.684i)8-s + (0.929 − 0.368i)9-s + (−0.248 − 0.968i)11-s + (−0.770 + 0.637i)12-s + (0.929 + 0.368i)14-s + (0.535 − 0.844i)16-s + (0.481 − 0.876i)17-s + (−0.809 + 0.587i)18-s + (0.982 + 0.187i)19-s + (0.904 + 0.425i)21-s + (0.481 + 0.876i)22-s + ⋯
L(s)  = 1  + (−0.968 + 0.248i)2-s + (−0.982 + 0.187i)3-s + (0.876 − 0.481i)4-s + (0.904 − 0.425i)6-s + (−0.809 − 0.587i)7-s + (−0.728 + 0.684i)8-s + (0.929 − 0.368i)9-s + (−0.248 − 0.968i)11-s + (−0.770 + 0.637i)12-s + (0.929 + 0.368i)14-s + (0.535 − 0.844i)16-s + (0.481 − 0.876i)17-s + (−0.809 + 0.587i)18-s + (0.982 + 0.187i)19-s + (0.904 + 0.425i)21-s + (0.481 + 0.876i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1625 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1625 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1625\)    =    \(5^{3} \cdot 13\)
Sign: $0.759 - 0.650i$
Analytic conductor: \(7.54646\)
Root analytic conductor: \(7.54646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1625} (1152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1625,\ (0:\ ),\ 0.759 - 0.650i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5889451161 - 0.2178868253i\)
\(L(\frac12)\) \(\approx\) \(0.5889451161 - 0.2178868253i\)
\(L(1)\) \(\approx\) \(0.5273564927 + 0.03006813810i\)
\(L(1)\) \(\approx\) \(0.5273564927 + 0.03006813810i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.968 + 0.248i)T \)
3 \( 1 + (-0.982 + 0.187i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-0.248 - 0.968i)T \)
17 \( 1 + (0.481 - 0.876i)T \)
19 \( 1 + (0.982 + 0.187i)T \)
23 \( 1 + (0.998 + 0.0627i)T \)
29 \( 1 + (0.992 + 0.125i)T \)
31 \( 1 + (-0.481 + 0.876i)T \)
37 \( 1 + (0.535 - 0.844i)T \)
41 \( 1 + (-0.998 + 0.0627i)T \)
43 \( 1 + (-0.951 + 0.309i)T \)
47 \( 1 + (0.728 + 0.684i)T \)
53 \( 1 + (0.904 + 0.425i)T \)
59 \( 1 + (0.770 - 0.637i)T \)
61 \( 1 + (0.0627 - 0.998i)T \)
67 \( 1 + (0.992 - 0.125i)T \)
71 \( 1 + (-0.684 + 0.728i)T \)
73 \( 1 + (0.637 - 0.770i)T \)
79 \( 1 + (0.187 + 0.982i)T \)
83 \( 1 + (-0.187 + 0.982i)T \)
89 \( 1 + (-0.770 - 0.637i)T \)
97 \( 1 + (0.992 + 0.125i)T \)
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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

−20.36883666183616320695827067985, −19.60648795651742386951511638350, −18.78952005639814215804157842696, −18.36336173984041112725863544985, −17.60666720657871903522615749315, −16.86620317364453465332629677833, −16.36414020109647424380256286828, −15.412609532052743487672218026479, −15.062222331804677760734306143968, −13.30269032613071949375976472429, −12.78966644904176170258647785886, −11.92824619133850090742753707503, −11.614569879436738136387631783190, −10.31074602331805099089396994971, −10.11372486457500398691405553606, −9.219202310377220970627615563250, −8.26398553313977223274144239650, −7.270620926761641210957317550634, −6.76074262145400105658573235184, −5.8899130005350766679019946756, −5.07648081952275632084009859175, −3.8141510733685175963516286370, −2.768414553496300896998438651101, −1.8234756357134477979982665076, −0.79795044146964951839246015653, 0.59014723199921590526093679671, 1.20243914477480521632678143909, 2.83758414858957258926771610743, 3.58081079328031698201531103574, 5.056041918916145090706647686492, 5.61287062272810427488908149977, 6.61107857179976219975192041224, 7.06178674122135273995190836388, 7.95629193110201097369994269330, 9.07436714899724735159082419617, 9.734360224789657124113474949703, 10.42542291958715249923830710122, 11.076585841654923102102548185973, 11.77936340292926907094349602125, 12.61418027264304374551293625076, 13.61896479064841327378561392732, 14.45806726359374767719466717399, 15.76817708892370532903217218414, 15.97161566954458630569556833725, 16.7077027689168233741282392449, 17.19849189753290650799048477078, 18.250099601420229276084464378467, 18.57414180170612395270006782920, 19.46653375626445672743780687568, 20.19981268701580799748248713008

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