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Properties

Label 1-4025-4025.2-r1-0-0
Degree $1$
Conductor $4025$
Sign $0.845 + 0.533i$
Analytic cond. $432.546$
Root an. cond. $432.546$
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  + (0.803 − 0.595i)2-s + (0.647 + 0.761i)3-s + (0.290 − 0.956i)4-s + (0.974 + 0.226i)6-s + (−0.336 − 0.941i)8-s + (−0.161 + 0.986i)9-s + (0.953 − 0.299i)11-s + (0.917 − 0.398i)12-s + (0.170 + 0.985i)13-s + (−0.830 − 0.556i)16-s + (−0.730 − 0.683i)17-s + (0.458 + 0.888i)18-s + (0.905 − 0.424i)19-s + (0.587 − 0.809i)22-s + (0.5 − 0.866i)24-s + ⋯
L(s)  = 1  + (0.803 − 0.595i)2-s + (0.647 + 0.761i)3-s + (0.290 − 0.956i)4-s + (0.974 + 0.226i)6-s + (−0.336 − 0.941i)8-s + (−0.161 + 0.986i)9-s + (0.953 − 0.299i)11-s + (0.917 − 0.398i)12-s + (0.170 + 0.985i)13-s + (−0.830 − 0.556i)16-s + (−0.730 − 0.683i)17-s + (0.458 + 0.888i)18-s + (0.905 − 0.424i)19-s + (0.587 − 0.809i)22-s + (0.5 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $0.845 + 0.533i$
Analytic conductor: \(432.546\)
Root analytic conductor: \(432.546\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4025} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4025,\ (1:\ ),\ 0.845 + 0.533i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.894467837 + 1.414641333i\)
\(L(\frac12)\) \(\approx\) \(4.894467837 + 1.414641333i\)
\(L(1)\) \(\approx\) \(2.114801412 - 0.1374353478i\)
\(L(1)\) \(\approx\) \(2.114801412 - 0.1374353478i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.803 - 0.595i)T \)
3 \( 1 + (0.647 + 0.761i)T \)
11 \( 1 + (0.953 - 0.299i)T \)
13 \( 1 + (0.170 + 0.985i)T \)
17 \( 1 + (-0.730 - 0.683i)T \)
19 \( 1 + (0.905 - 0.424i)T \)
29 \( 1 + (-0.0855 - 0.996i)T \)
31 \( 1 + (-0.179 + 0.983i)T \)
37 \( 1 + (0.986 + 0.161i)T \)
41 \( 1 + (-0.254 + 0.967i)T \)
43 \( 1 + (0.281 + 0.959i)T \)
47 \( 1 + (-0.994 - 0.104i)T \)
53 \( 1 + (-0.846 + 0.532i)T \)
59 \( 1 + (-0.345 + 0.938i)T \)
61 \( 1 + (0.997 + 0.0760i)T \)
67 \( 1 + (0.508 + 0.861i)T \)
71 \( 1 + (0.198 - 0.980i)T \)
73 \( 1 + (0.0190 + 0.999i)T \)
79 \( 1 + (0.969 + 0.244i)T \)
83 \( 1 + (0.931 + 0.362i)T \)
89 \( 1 + (-0.879 + 0.475i)T \)
97 \( 1 + (0.931 - 0.362i)T \)
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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

−18.04903850026475522871295269576, −17.56209284602088054853442403513, −16.93375412975014017469642741699, −16.0150031873455585549478586491, −15.30354767619254899664085267098, −14.73456585844141668712360893456, −14.213238418608880239327564836565, −13.495321588500779546868110884774, −12.84325030707438036775381724126, −12.40905403226231799606700491890, −11.63238690229257420007814409639, −10.91623873620282643557822732942, −9.70593471938209351360951410644, −8.982279964425909585652456563629, −8.24623366183280423621730674069, −7.68088002347413690586276930008, −6.9636529235349453866063612686, −6.3207049100199696574400979756, −5.68159392635364351778282939027, −4.75168923870400758230417624756, −3.65994240518893658024477030625, −3.45761994875722052490525479438, −2.330941522081605184689742867626, −1.65487011054935442381468069258, −0.50708216668170850618587495047, 0.92531993087587859744815052254, 1.78759077861135313980889008007, 2.67955642345015329503788208279, 3.27380199864997167803278710378, 4.15415732607126316699022086080, 4.557894629078378775437550573066, 5.34464092600795480786976799322, 6.34028891728887323017531287028, 6.92049293920157679808256928276, 7.98276701672638978075602419749, 9.014050749356198977857988040069, 9.43337487811849698088375596287, 9.95890073777240569138058140089, 11.135429010616573976778057230991, 11.32562205242970147833396825708, 12.04410406058083077542581087833, 13.16644246371295845221988748518, 13.65371424719109205918587957216, 14.24855706739325046680039901180, 14.73816524223270442650925303965, 15.54723712900131287548642182691, 16.1623633282399466441174121459, 16.65964718954574406584981807195, 17.81374934965882308782747843829, 18.60618356866218427660401493139

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