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Properties

Label 1-35e2-1225.4-r0-0-0
Degree $1$
Conductor $1225$
Sign $0.593 + 0.804i$
Analytic cond. $5.68887$
Root an. cond. $5.68887$
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  + (−0.251 + 0.967i)2-s + (0.925 − 0.379i)3-s + (−0.873 − 0.486i)4-s + (0.134 + 0.990i)6-s + (0.691 − 0.722i)8-s + (0.712 − 0.701i)9-s + (0.712 + 0.701i)11-s + (−0.992 − 0.119i)12-s + (0.0448 − 0.998i)13-s + (0.525 + 0.850i)16-s + (−0.0149 + 0.999i)17-s + (0.5 + 0.866i)18-s + (0.669 + 0.743i)19-s + (−0.858 + 0.512i)22-s + (0.599 + 0.800i)23-s + (0.365 − 0.930i)24-s + ⋯
L(s)  = 1  + (−0.251 + 0.967i)2-s + (0.925 − 0.379i)3-s + (−0.873 − 0.486i)4-s + (0.134 + 0.990i)6-s + (0.691 − 0.722i)8-s + (0.712 − 0.701i)9-s + (0.712 + 0.701i)11-s + (−0.992 − 0.119i)12-s + (0.0448 − 0.998i)13-s + (0.525 + 0.850i)16-s + (−0.0149 + 0.999i)17-s + (0.5 + 0.866i)18-s + (0.669 + 0.743i)19-s + (−0.858 + 0.512i)22-s + (0.599 + 0.800i)23-s + (0.365 − 0.930i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.593 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.593 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.593 + 0.804i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ 0.593 + 0.804i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.692641233 + 0.8547119376i\)
\(L(\frac12)\) \(\approx\) \(1.692641233 + 0.8547119376i\)
\(L(1)\) \(\approx\) \(1.223293860 + 0.4201422683i\)
\(L(1)\) \(\approx\) \(1.223293860 + 0.4201422683i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.251 + 0.967i)T \)
3 \( 1 + (0.925 - 0.379i)T \)
11 \( 1 + (0.712 + 0.701i)T \)
13 \( 1 + (0.0448 - 0.998i)T \)
17 \( 1 + (-0.0149 + 0.999i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (0.599 + 0.800i)T \)
29 \( 1 + (-0.995 + 0.0896i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (-0.992 - 0.119i)T \)
41 \( 1 + (0.983 - 0.178i)T \)
43 \( 1 + (0.900 - 0.433i)T \)
47 \( 1 + (0.772 - 0.635i)T \)
53 \( 1 + (0.873 + 0.486i)T \)
59 \( 1 + (-0.999 + 0.0299i)T \)
61 \( 1 + (0.193 + 0.981i)T \)
67 \( 1 + (0.978 - 0.207i)T \)
71 \( 1 + (0.858 - 0.512i)T \)
73 \( 1 + (0.842 - 0.538i)T \)
79 \( 1 + (-0.978 - 0.207i)T \)
83 \( 1 + (-0.936 + 0.351i)T \)
89 \( 1 + (0.887 + 0.460i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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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

−20.888585055947112031095411621433, −20.262168030331692675526685923329, −19.59128408908528428541797307314, −18.824541313896472322489126281087, −18.4229567386915929434965703892, −17.18050097784464377142223225267, −16.47383956581921364246513002996, −15.7072222769166215721894076450, −14.35740957841138786880271238818, −14.1458022672711864458066912510, −13.28919503352281939466150601392, −12.46445615981577329662919924974, −11.33153078469899284770490711160, −11.02596939126167660812815762635, −9.78305283129283690013082227278, −9.13947607668837937477819661072, −8.833054777792982104015251616223, −7.67290771746096388606578019250, −6.90183995664256250441104247640, −5.351231433590301203595724503691, −4.42434561784111506730046761990, −3.70538347543808367851752557578, −2.86138366337468177453452344990, −2.05248490710857255390586800666, −0.9341124132071127360674883848, 1.08896033802816569526275792111, 1.95261908539983485043206525095, 3.488274574767721531544027937210, 4.00929427610784374455335672049, 5.33821312623277787284424199009, 6.07329412741990393994394966226, 7.30194846716360398404158023809, 7.463239010948019071785909088797, 8.534002026235149922702863537789, 9.17645807837835481085759691618, 9.904936793990489593110844091378, 10.78483558499152523838308924262, 12.332726567066052397701025426078, 12.81686463402633764867900877806, 13.74090337225707524040801442323, 14.42251478341279070914477607975, 15.14504777471970476710601983201, 15.546779236283756213050619667679, 16.71457263683784948905811066177, 17.4520694495653747369041340272, 18.11839640019914783448900931155, 18.87602322303794375445848821445, 19.66317816699859282220305251027, 20.1999224840836360845068043797, 21.18375500393482881453575853394

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