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Properties

Label 1-4025-4025.867-r0-0-0
Degree $1$
Conductor $4025$
Sign $-0.999 + 0.0346i$
Analytic cond. $18.6920$
Root an. cond. $18.6920$
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  + (−0.717 + 0.696i)2-s + (0.676 − 0.736i)3-s + (0.0285 − 0.999i)4-s + (0.0285 + 0.999i)6-s + (0.676 + 0.736i)8-s + (−0.0855 − 0.996i)9-s + (−0.466 − 0.884i)11-s + (−0.717 − 0.696i)12-s + (0.931 − 0.362i)13-s + (−0.998 − 0.0570i)16-s + (−0.999 + 0.0285i)17-s + (0.755 + 0.654i)18-s + (−0.564 − 0.825i)19-s + (0.951 + 0.309i)22-s + 24-s + ⋯
L(s)  = 1  + (−0.717 + 0.696i)2-s + (0.676 − 0.736i)3-s + (0.0285 − 0.999i)4-s + (0.0285 + 0.999i)6-s + (0.676 + 0.736i)8-s + (−0.0855 − 0.996i)9-s + (−0.466 − 0.884i)11-s + (−0.717 − 0.696i)12-s + (0.931 − 0.362i)13-s + (−0.998 − 0.0570i)16-s + (−0.999 + 0.0285i)17-s + (0.755 + 0.654i)18-s + (−0.564 − 0.825i)19-s + (0.951 + 0.309i)22-s + 24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0346i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0346i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $-0.999 + 0.0346i$
Analytic conductor: \(18.6920\)
Root analytic conductor: \(18.6920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4025} (867, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4025,\ (0:\ ),\ -0.999 + 0.0346i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.009894976438 - 0.5715040607i\)
\(L(\frac12)\) \(\approx\) \(0.009894976438 - 0.5715040607i\)
\(L(1)\) \(\approx\) \(0.7545073166 - 0.1881454844i\)
\(L(1)\) \(\approx\) \(0.7545073166 - 0.1881454844i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.717 + 0.696i)T \)
3 \( 1 + (0.676 - 0.736i)T \)
11 \( 1 + (-0.466 - 0.884i)T \)
13 \( 1 + (0.931 - 0.362i)T \)
17 \( 1 + (-0.999 + 0.0285i)T \)
19 \( 1 + (-0.564 - 0.825i)T \)
29 \( 1 + (0.564 - 0.825i)T \)
31 \( 1 + (0.736 - 0.676i)T \)
37 \( 1 + (-0.996 + 0.0855i)T \)
41 \( 1 + (-0.974 + 0.226i)T \)
43 \( 1 + (-0.909 - 0.415i)T \)
47 \( 1 + (-0.951 - 0.309i)T \)
53 \( 1 + (-0.967 + 0.254i)T \)
59 \( 1 + (-0.362 - 0.931i)T \)
61 \( 1 + (0.870 + 0.491i)T \)
67 \( 1 + (0.441 + 0.897i)T \)
71 \( 1 + (-0.985 - 0.170i)T \)
73 \( 1 + (0.791 + 0.610i)T \)
79 \( 1 + (-0.774 + 0.633i)T \)
83 \( 1 + (0.856 + 0.516i)T \)
89 \( 1 + (0.198 + 0.980i)T \)
97 \( 1 + (0.856 - 0.516i)T \)
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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

−18.95069324102435447000750338249, −18.184939411325000528148646935818, −17.599352620931753174188898802174, −16.79736318150877387859190064138, −16.02516986942996578823796830808, −15.66363884872959212797221448046, −14.81398802651833997899020917246, −13.94643507126639928966772683689, −13.30571526604202673640634075809, −12.64037498955407411273189958688, −11.81497776069140413014533145687, −10.99177794873737126601373464777, −10.405853659212090707499436616488, −9.95548278609344640172819235681, −9.08241963671439253281270892751, −8.53718767405749402721312744150, −8.05294251605841152305216478185, −7.08294058850069954144504948783, −6.37683679417632275718775352540, −4.93133714472025959306178482266, −4.4911449544601025296513549502, −3.56496284776689198325671153279, −3.04022369380945382379361756867, −1.970266729792946390805472898310, −1.6010802567138773076404646899, 0.18887693291106338741286487034, 1.05443754846071064193507707857, 2.01153387408862172540520154735, 2.74504372577358709050177669433, 3.7063003690572242140624658965, 4.76050829815488580451271418828, 5.68183917819952922899902888332, 6.57571524500195542653520635722, 6.69724160704513411272943257334, 7.897238858464462758813398856156, 8.3751272362879748495826286504, 8.72602588874039882272374884955, 9.60202039646243052804422088889, 10.4242411731761723338178063162, 11.17803920206315872573792081611, 11.78478927211894562124775498246, 13.13342244360058368280076265919, 13.379179667908526485517230320170, 13.98260687182112481379658577147, 14.873513904474437017486597292419, 15.595054322687174510138315168501, 15.81858593704468154316176615091, 16.97511321528638577805523690009, 17.54221659129511715183396226385, 18.12430265054822489172648594641

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