↑

Properties

Label 1-925-925.309-r1-0-0
Degree $1$
Conductor $925$
Sign $0.162 + 0.986i$
Analytic cond. $99.4050$
Root an. cond. $99.4050$
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  + (0.999 + 0.0348i)2-s + (0.559 − 0.829i)3-s + (0.997 + 0.0697i)4-s + (0.587 − 0.809i)6-s + (−0.173 + 0.984i)7-s + (0.994 + 0.104i)8-s + (−0.374 − 0.927i)9-s + (−0.669 + 0.743i)11-s + (0.615 − 0.788i)12-s + (−0.529 − 0.848i)13-s + (−0.207 + 0.978i)14-s + (0.990 + 0.139i)16-s + (0.0697 + 0.997i)17-s + (−0.342 − 0.939i)18-s + (−0.275 + 0.961i)19-s + ⋯
L(s)  = 1  + (0.999 + 0.0348i)2-s + (0.559 − 0.829i)3-s + (0.997 + 0.0697i)4-s + (0.587 − 0.809i)6-s + (−0.173 + 0.984i)7-s + (0.994 + 0.104i)8-s + (−0.374 − 0.927i)9-s + (−0.669 + 0.743i)11-s + (0.615 − 0.788i)12-s + (−0.529 − 0.848i)13-s + (−0.207 + 0.978i)14-s + (0.990 + 0.139i)16-s + (0.0697 + 0.997i)17-s + (−0.342 − 0.939i)18-s + (−0.275 + 0.961i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.162 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.162 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $0.162 + 0.986i$
Analytic conductor: \(99.4050\)
Root analytic conductor: \(99.4050\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 925,\ (1:\ ),\ 0.162 + 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.653967094 + 2.252080917i\)
\(L(\frac12)\) \(\approx\) \(2.653967094 + 2.252080917i\)
\(L(1)\) \(\approx\) \(2.014817045 + 0.1020230538i\)
\(L(1)\) \(\approx\) \(2.014817045 + 0.1020230538i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.999 + 0.0348i)T \)
3 \( 1 + (0.559 - 0.829i)T \)
7 \( 1 + (-0.173 + 0.984i)T \)
11 \( 1 + (-0.669 + 0.743i)T \)
13 \( 1 + (-0.529 - 0.848i)T \)
17 \( 1 + (0.0697 + 0.997i)T \)
19 \( 1 + (-0.275 + 0.961i)T \)
23 \( 1 + (-0.207 + 0.978i)T \)
29 \( 1 + (0.406 - 0.913i)T \)
31 \( 1 + (-0.587 + 0.809i)T \)
41 \( 1 + (-0.848 + 0.529i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.104 + 0.994i)T \)
53 \( 1 + (0.719 + 0.694i)T \)
59 \( 1 + (-0.139 + 0.990i)T \)
61 \( 1 + (0.927 + 0.374i)T \)
67 \( 1 + (-0.719 + 0.694i)T \)
71 \( 1 + (0.559 - 0.829i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.898 + 0.438i)T \)
83 \( 1 + (0.997 - 0.0697i)T \)
89 \( 1 + (0.469 + 0.882i)T \)
97 \( 1 + (-0.994 + 0.104i)T \)
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

−21.563439835705349967870763194947, −20.765434480522621911215862824003, −20.171093522817812513786803738156, −19.52262866556048406193882460738, −18.60379826919728502780027495281, −17.0979850097504445552824561160, −16.34542305890437102227644018521, −16.014883028165842232656311620402, −14.91663454026675116910771890578, −14.24812447855780109308726827514, −13.61971223993411776171949085047, −13.00154423723478435819451352109, −11.71329029178385528831997899348, −10.949939503219431388820742090295, −10.32322132518292600073929653101, −9.39945130634493078403696628130, −8.28849791648192913939332734197, −7.30698723105060282613299802139, −6.57326801857928156415254764443, −5.20709308465621003193725621285, −4.67300461636184050829365171249, −3.780522713653243867770760706824, −2.95618133175678583094330959470, −2.1167864951202688171533046873, −0.40829742017965892847186106395, 1.49560158808742421856228674811, 2.30255383495467396544231538298, 3.04176046501512759357847178266, 4.03010509517615041636248876526, 5.39149119834735257996761323124, 5.88079480658237945743084992244, 6.91587323291749165803198450353, 7.79815739563639132491570803013, 8.38037595099151995017093607241, 9.71910739936003752444998281128, 10.57006097132406478361747611916, 11.922761998604241035428723377224, 12.34598904808610861707261885762, 12.95527562852316641422288798668, 13.714977480391415754051838403699, 14.86966540345340553269666786875, 15.05635168931759090545514082799, 15.91809968765102681319095009863, 17.192163126936736160031708155249, 17.92685175392976720945256806985, 18.911150680707788306036850807682, 19.57706101059120476371085173154, 20.32117686789722235796996117995, 21.11675271349089890111406897648, 21.82156156114745865175902212310

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