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Properties

Label 1-1625-1625.1134-r0-0-0
Degree $1$
Conductor $1625$
Sign $0.455 - 0.889i$
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  + (−0.985 + 0.166i)2-s + (−0.387 + 0.921i)3-s + (0.944 − 0.328i)4-s + (0.228 − 0.973i)6-s + (−0.913 − 0.406i)7-s + (−0.876 + 0.481i)8-s + (−0.699 − 0.714i)9-s + (0.985 − 0.166i)11-s + (−0.0627 + 0.998i)12-s + (0.968 + 0.248i)14-s + (0.783 − 0.621i)16-s + (0.756 − 0.653i)17-s + (0.809 + 0.587i)18-s + (0.387 + 0.921i)19-s + (0.728 − 0.684i)21-s + (−0.944 + 0.328i)22-s + ⋯
L(s)  = 1  + (−0.985 + 0.166i)2-s + (−0.387 + 0.921i)3-s + (0.944 − 0.328i)4-s + (0.228 − 0.973i)6-s + (−0.913 − 0.406i)7-s + (−0.876 + 0.481i)8-s + (−0.699 − 0.714i)9-s + (0.985 − 0.166i)11-s + (−0.0627 + 0.998i)12-s + (0.968 + 0.248i)14-s + (0.783 − 0.621i)16-s + (0.756 − 0.653i)17-s + (0.809 + 0.587i)18-s + (0.387 + 0.921i)19-s + (0.728 − 0.684i)21-s + (−0.944 + 0.328i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1625 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.455 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1625 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.455 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1625\)    =    \(5^{3} \cdot 13\)
Sign: $0.455 - 0.889i$
Analytic conductor: \(7.54646\)
Root analytic conductor: \(7.54646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1625} (1134, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1625,\ (0:\ ),\ 0.455 - 0.889i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4126732807 - 0.2522559929i\)
\(L(\frac12)\) \(\approx\) \(0.4126732807 - 0.2522559929i\)
\(L(1)\) \(\approx\) \(0.5394856236 + 0.07826737199i\)
\(L(1)\) \(\approx\) \(0.5394856236 + 0.07826737199i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.985 + 0.166i)T \)
3 \( 1 + (-0.387 + 0.921i)T \)
7 \( 1 + (-0.913 - 0.406i)T \)
11 \( 1 + (0.985 - 0.166i)T \)
17 \( 1 + (0.756 - 0.653i)T \)
19 \( 1 + (0.387 + 0.921i)T \)
23 \( 1 + (-0.463 - 0.886i)T \)
29 \( 1 + (0.996 + 0.0836i)T \)
31 \( 1 + (-0.187 - 0.982i)T \)
37 \( 1 + (-0.783 + 0.621i)T \)
41 \( 1 + (-0.999 + 0.0418i)T \)
43 \( 1 + (0.978 - 0.207i)T \)
47 \( 1 + (-0.876 - 0.481i)T \)
53 \( 1 + (-0.728 + 0.684i)T \)
59 \( 1 + (-0.895 + 0.444i)T \)
61 \( 1 + (-0.999 - 0.0418i)T \)
67 \( 1 + (0.570 + 0.821i)T \)
71 \( 1 + (-0.0209 - 0.999i)T \)
73 \( 1 + (-0.0627 - 0.998i)T \)
79 \( 1 + (-0.992 + 0.125i)T \)
83 \( 1 + (0.992 + 0.125i)T \)
89 \( 1 + (-0.895 - 0.444i)T \)
97 \( 1 + (0.570 - 0.821i)T \)
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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

−19.99421837414500981136171073798, −19.667620187226621163513918468, −19.10467877316384358135256539423, −18.406389280661018514444421828526, −17.46112247836047070391119198434, −17.27288743968150453043989972750, −16.18670418821300414045831547485, −15.73978706662028729235620043320, −14.58865096990120138148641250973, −13.70469760759820227471631777171, −12.655973134805528021221454443178, −12.22293399280132804675038117885, −11.56655494661676589604544067337, −10.70214612463902513133700574848, −9.795050044914874096006767512891, −9.09593124279443909621623211956, −8.313489868956307940872589947519, −7.43140351872167355340537790625, −6.671751248387625606746346564334, −6.20676288206085940598591653319, −5.23177875637851463822190678655, −3.6046537060894048412370932649, −2.86506914814593114186320717933, −1.79492213086467856130285090871, −1.050467918090363945611682572381, 0.30794083558701190411054863112, 1.40064904832955508781873268930, 2.91188084217032487367702259614, 3.536610234738255218915169045418, 4.58506157593720918269958609689, 5.79665697482143023895875526879, 6.30612998525639238608494097405, 7.12480860256799606400114280188, 8.175905739539269815598034773845, 9.03321889384032913773161595814, 9.7400974639862433622514424537, 10.16267570308177670496326778699, 10.93711114910748712270608236087, 11.96434247535834613193562647528, 12.246707123979136932143376445955, 13.87377460162994811833739689949, 14.46573153000072582831927879897, 15.39378487043397597152905658945, 16.08683767173470633874923275600, 16.74156663361582482058447719959, 16.95846168869604159096058546833, 18.04013917840410966012392748607, 18.794221839720228568692476234208, 19.54701679750450836776424830069, 20.35039574892532134880425033117

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