↑

Properties

Label 1-1625-1625.1109-r0-0-0
Degree $1$
Conductor $1625$
Sign $-0.814 + 0.580i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.268 − 0.963i)2-s + (−0.944 + 0.328i)3-s + (−0.855 + 0.518i)4-s + (0.570 + 0.821i)6-s + (−0.104 − 0.994i)7-s + (0.728 + 0.684i)8-s + (0.783 − 0.621i)9-s + (0.268 + 0.963i)11-s + (0.637 − 0.770i)12-s + (−0.929 + 0.368i)14-s + (0.463 − 0.886i)16-s + (0.0209 − 0.999i)17-s + (−0.809 − 0.587i)18-s + (−0.944 − 0.328i)19-s + (0.425 + 0.904i)21-s + (0.855 − 0.518i)22-s + ⋯
L(s)  = 1  + (−0.268 − 0.963i)2-s + (−0.944 + 0.328i)3-s + (−0.855 + 0.518i)4-s + (0.570 + 0.821i)6-s + (−0.104 − 0.994i)7-s + (0.728 + 0.684i)8-s + (0.783 − 0.621i)9-s + (0.268 + 0.963i)11-s + (0.637 − 0.770i)12-s + (−0.929 + 0.368i)14-s + (0.463 − 0.886i)16-s + (0.0209 − 0.999i)17-s + (−0.809 − 0.587i)18-s + (−0.944 − 0.328i)19-s + (0.425 + 0.904i)21-s + (0.855 − 0.518i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1625 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1625 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1625\)    =    \(5^{3} \cdot 13\)
Sign: $-0.814 + 0.580i$
Analytic conductor: \(7.54646\)
Root analytic conductor: \(7.54646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1625} (1109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1625,\ (0:\ ),\ -0.814 + 0.580i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06083165980 - 0.1900080263i\)
\(L(\frac12)\) \(\approx\) \(-0.06083165980 - 0.1900080263i\)
\(L(1)\) \(\approx\) \(0.4874186619 - 0.2403292628i\)
\(L(1)\) \(\approx\) \(0.4874186619 - 0.2403292628i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.268 - 0.963i)T \)
3 \( 1 + (-0.944 + 0.328i)T \)
7 \( 1 + (-0.104 - 0.994i)T \)
11 \( 1 + (0.268 + 0.963i)T \)
17 \( 1 + (0.0209 - 0.999i)T \)
19 \( 1 + (-0.944 - 0.328i)T \)
23 \( 1 + (-0.832 + 0.553i)T \)
29 \( 1 + (0.604 + 0.796i)T \)
31 \( 1 + (-0.876 + 0.481i)T \)
37 \( 1 + (0.463 - 0.886i)T \)
41 \( 1 + (0.895 - 0.444i)T \)
43 \( 1 + (-0.669 - 0.743i)T \)
47 \( 1 + (0.728 - 0.684i)T \)
53 \( 1 + (0.425 + 0.904i)T \)
59 \( 1 + (0.348 + 0.937i)T \)
61 \( 1 + (-0.895 - 0.444i)T \)
67 \( 1 + (0.387 + 0.921i)T \)
71 \( 1 + (-0.228 - 0.973i)T \)
73 \( 1 + (-0.637 - 0.770i)T \)
79 \( 1 + (-0.187 + 0.982i)T \)
83 \( 1 + (-0.187 - 0.982i)T \)
89 \( 1 + (0.348 - 0.937i)T \)
97 \( 1 + (0.387 - 0.921i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.31682752492910513467660648724, −19.72824642520594095655605216819, −18.97341288841431653076569643403, −18.6195605135314901803583162128, −17.80575504955254858615545383125, −17.10083261965272912746728636724, −16.4531065729092551047523119152, −15.88380040199989902958421595764, −15.02311711072273070152295869137, −14.354620622606765136693378180617, −13.26847035651144983921817113672, −12.73494281357826213893026590628, −11.85963853894491114048427689392, −10.98517845204102094025356939910, −10.2008704612943288500362599951, −9.33430208380400071745018932759, −8.296673454020170627092480463150, −7.9902370859416602787360855773, −6.618750728475445009482296862883, −6.125073944444947123571570227973, −5.721576816941475975707023079599, −4.668039510363849858287932061146, −3.844314763803050179980326231414, −2.27593322398034703000094514418, −1.1917170031469167319042607315, 0.110217350717183768040530959778, 1.18177796819947567517176671113, 2.15307419136639924689336700985, 3.45834846343152827813612237819, 4.24720466366997413110968636014, 4.74745386048561888172138726847, 5.781308872221350378490660898400, 7.08287008934652691701323340685, 7.40056412035935950233068306882, 8.84333991784853368643709241921, 9.54082372662054867858377952501, 10.33813299510772254971321971048, 10.72757000355616809739021638667, 11.618585012661612962038282303445, 12.283499490551713467604450908447, 12.93096020125721290402818299570, 13.81097231792642696275369507356, 14.620846742672005788940754914863, 15.74112928840142098570871660943, 16.55982460440036565761471401149, 17.13735776821427902120106731624, 17.88684692025006028797798297878, 18.25486964932103234541241489216, 19.43910777511874194974053875422, 20.07666443365681977985241504173

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