Properties

Label 1-4025-4025.1553-r0-0-0
Degree $1$
Conductor $4025$
Sign $0.0176 - 0.999i$
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  + (0.491 + 0.870i)2-s + (−0.999 + 0.0285i)3-s + (−0.516 + 0.856i)4-s + (−0.516 − 0.856i)6-s + (−0.999 − 0.0285i)8-s + (0.998 − 0.0570i)9-s + (0.198 − 0.980i)11-s + (0.491 − 0.870i)12-s + (0.717 − 0.696i)13-s + (−0.466 − 0.884i)16-s + (−0.856 + 0.516i)17-s + (0.540 + 0.841i)18-s + (−0.921 + 0.389i)19-s + (0.951 − 0.309i)22-s + 24-s + ⋯
L(s)  = 1  + (0.491 + 0.870i)2-s + (−0.999 + 0.0285i)3-s + (−0.516 + 0.856i)4-s + (−0.516 − 0.856i)6-s + (−0.999 − 0.0285i)8-s + (0.998 − 0.0570i)9-s + (0.198 − 0.980i)11-s + (0.491 − 0.870i)12-s + (0.717 − 0.696i)13-s + (−0.466 − 0.884i)16-s + (−0.856 + 0.516i)17-s + (0.540 + 0.841i)18-s + (−0.921 + 0.389i)19-s + (0.951 − 0.309i)22-s + 24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0176 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0176 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $0.0176 - 0.999i$
Analytic conductor: \(18.6920\)
Root analytic conductor: \(18.6920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4025} (1553, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4025,\ (0:\ ),\ 0.0176 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2031075570 - 0.2067215800i\)
\(L(\frac12)\) \(\approx\) \(0.2031075570 - 0.2067215800i\)
\(L(1)\) \(\approx\) \(0.7351805813 + 0.3126666600i\)
\(L(1)\) \(\approx\) \(0.7351805813 + 0.3126666600i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.491 + 0.870i)T \)
3 \( 1 + (-0.999 + 0.0285i)T \)
11 \( 1 + (0.198 - 0.980i)T \)
13 \( 1 + (0.717 - 0.696i)T \)
17 \( 1 + (-0.856 + 0.516i)T \)
19 \( 1 + (-0.921 + 0.389i)T \)
29 \( 1 + (0.921 + 0.389i)T \)
31 \( 1 + (0.0285 - 0.999i)T \)
37 \( 1 + (0.0570 + 0.998i)T \)
41 \( 1 + (0.362 - 0.931i)T \)
43 \( 1 + (0.281 - 0.959i)T \)
47 \( 1 + (-0.951 + 0.309i)T \)
53 \( 1 + (-0.170 - 0.985i)T \)
59 \( 1 + (0.696 + 0.717i)T \)
61 \( 1 + (-0.941 - 0.336i)T \)
67 \( 1 + (-0.676 + 0.736i)T \)
71 \( 1 + (0.993 + 0.113i)T \)
73 \( 1 + (0.996 - 0.0855i)T \)
79 \( 1 + (-0.897 + 0.441i)T \)
83 \( 1 + (-0.633 - 0.774i)T \)
89 \( 1 + (0.610 + 0.791i)T \)
97 \( 1 + (-0.633 + 0.774i)T \)
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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

−18.55886696519263199922287287748, −17.93992750283594218391482064332, −17.61405502316425319966358302624, −16.63798874260352169090170784591, −15.82178923481276514452490940548, −15.300641350253085631005895888580, −14.42593460217793349100482065036, −13.68609993068497325954927083646, −12.94263335658710449129341648734, −12.45341500332263991666803627398, −11.75523622920724940200103388241, −11.09833509585982915550090251023, −10.69630037798799350616910767279, −9.7791727738701469026201246833, −9.26800192093835728657150104656, −8.37468592983626603438814472367, −7.099415759148972808235911632301, −6.503070133268502676382722304767, −5.96367821549247314293430716904, −4.742064298960187551552291325972, −4.631903940467706185488259827876, −3.82583420282451171459608870892, −2.66004141317518442371712940540, −1.84328617874206244890214002030, −1.138726992107656238100135666691, 0.08982219084211588747814555207, 1.16670334826363938110104751852, 2.48438082721505047472072650389, 3.63577733972819787770549001059, 4.09056997995890738634453174866, 5.00452913020425038546759070232, 5.68391688045081862216062210032, 6.35623199165537607864386167009, 6.644637356455762220826625275677, 7.77648969766524791532927932899, 8.40373757060581029342907041382, 9.03853476658403225176901708268, 10.15948499983510632578596443093, 10.82598655107922998353145009515, 11.498065285817814361506089552831, 12.234128130694185482642398299029, 13.02823181977763707052841219612, 13.35660232860937746805302213052, 14.247826794228881123545632342522, 15.116410957232120869369414196639, 15.66790649743148864619186704460, 16.21502936650516212026518126664, 16.962267608578783984027727705386, 17.364527146792348676839555006226, 18.10035488772570301483406034639

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