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Properties

Label 1-35e2-1225.946-r0-0-0
Degree $1$
Conductor $1225$
Sign $0.873 + 0.486i$
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  + (0.473 + 0.880i)2-s + (0.858 − 0.512i)3-s + (−0.550 + 0.834i)4-s + (0.858 + 0.512i)6-s + (−0.995 − 0.0896i)8-s + (0.473 − 0.880i)9-s + (0.473 + 0.880i)11-s + (−0.0448 + 0.998i)12-s + (0.983 − 0.178i)13-s + (−0.393 − 0.919i)16-s + (−0.550 − 0.834i)17-s + 18-s + (0.309 − 0.951i)19-s + (−0.550 + 0.834i)22-s + (−0.0448 − 0.998i)23-s + (−0.900 + 0.433i)24-s + ⋯
L(s)  = 1  + (0.473 + 0.880i)2-s + (0.858 − 0.512i)3-s + (−0.550 + 0.834i)4-s + (0.858 + 0.512i)6-s + (−0.995 − 0.0896i)8-s + (0.473 − 0.880i)9-s + (0.473 + 0.880i)11-s + (−0.0448 + 0.998i)12-s + (0.983 − 0.178i)13-s + (−0.393 − 0.919i)16-s + (−0.550 − 0.834i)17-s + 18-s + (0.309 − 0.951i)19-s + (−0.550 + 0.834i)22-s + (−0.0448 − 0.998i)23-s + (−0.900 + 0.433i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.873 + 0.486i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (946, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ 0.873 + 0.486i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.598143215 + 0.6741968570i\)
\(L(\frac12)\) \(\approx\) \(2.598143215 + 0.6741968570i\)
\(L(1)\) \(\approx\) \(1.699358204 + 0.4722285061i\)
\(L(1)\) \(\approx\) \(1.699358204 + 0.4722285061i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.473 + 0.880i)T \)
3 \( 1 + (0.858 - 0.512i)T \)
11 \( 1 + (0.473 + 0.880i)T \)
13 \( 1 + (0.983 - 0.178i)T \)
17 \( 1 + (-0.550 - 0.834i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.0448 - 0.998i)T \)
29 \( 1 + (0.936 + 0.351i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (-0.0448 + 0.998i)T \)
41 \( 1 + (0.753 + 0.657i)T \)
43 \( 1 + (-0.222 + 0.974i)T \)
47 \( 1 + (0.134 - 0.990i)T \)
53 \( 1 + (-0.550 + 0.834i)T \)
59 \( 1 + (-0.393 - 0.919i)T \)
61 \( 1 + (-0.963 + 0.266i)T \)
67 \( 1 + (0.309 - 0.951i)T \)
71 \( 1 + (-0.550 + 0.834i)T \)
73 \( 1 + (0.983 + 0.178i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (0.134 + 0.990i)T \)
89 \( 1 + (0.983 + 0.178i)T \)
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

−21.17482127747818856930675333173, −20.41762516804513223582979525326, −19.431522313252630781461533685370, −19.29990107380852536940042580251, −18.302769788679794687833757724688, −17.366866102083432970220523701687, −16.102520716737605954392376583910, −15.63609020796708060764994872102, −14.61173586380018324819617888130, −13.96689847047958777167033715488, −13.51529809387298745063041339325, −12.57720108238817269773828286809, −11.634575272763738661645953520628, −10.7691258345986922975028615870, −10.28060914598363696737971274497, −9.144771108959450647953844121929, −8.76460494912993342273018024944, −7.82194822504516042669320471476, −6.3585736281894719088987347607, −5.63063414105993899828682663211, −4.48937330260091589287120354805, −3.68824303752035336512050589138, −3.2579018443962972679751092777, −2.03634179417974897366138170088, −1.22990080919343787144605757790, 0.963333259498090373563949776808, 2.41147240947644404349656660685, 3.16370552919972009062124183263, 4.24918001367870109466706823737, 4.85922891944373750336064355186, 6.34443844549906506887192660762, 6.68291903661132530752924464587, 7.61113013217241853361104969610, 8.3599475021111047875118804709, 9.10816344759382741262775615410, 9.77623671412421940274174702625, 11.25938706796527669866394234290, 12.17642112273950370524573585601, 12.89848351134602897623105919456, 13.63478102313183729269059728513, 14.136658076802405470667547799271, 15.11638519902434156605009750052, 15.510050625809165526709868866496, 16.41884127876540916149952756882, 17.447795347250016673199054651595, 18.12693561941559706028891994427, 18.607879328898851027180177036554, 19.882820744863927154975962882670, 20.38082832050043650074655648492, 21.17086514586671541339606144068

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