↑

Properties

Label 1-23e2-529.300-r0-0-0
Degree $1$
Conductor $529$
Sign $0.168 + 0.985i$
Analytic cond. $2.45666$
Root an. cond. $2.45666$
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  + (0.203 + 0.979i)2-s + (0.203 + 0.979i)3-s + (−0.917 + 0.398i)4-s + (0.460 − 0.887i)5-s + (−0.917 + 0.398i)6-s + (−0.917 − 0.398i)7-s + (−0.576 − 0.816i)8-s + (−0.917 + 0.398i)9-s + (0.962 + 0.269i)10-s + (0.854 − 0.519i)11-s + (−0.576 − 0.816i)12-s + (0.962 − 0.269i)13-s + (0.203 − 0.979i)14-s + (0.962 + 0.269i)15-s + (0.682 − 0.730i)16-s + (0.962 − 0.269i)17-s + ⋯
L(s)  = 1  + (0.203 + 0.979i)2-s + (0.203 + 0.979i)3-s + (−0.917 + 0.398i)4-s + (0.460 − 0.887i)5-s + (−0.917 + 0.398i)6-s + (−0.917 − 0.398i)7-s + (−0.576 − 0.816i)8-s + (−0.917 + 0.398i)9-s + (0.962 + 0.269i)10-s + (0.854 − 0.519i)11-s + (−0.576 − 0.816i)12-s + (0.962 − 0.269i)13-s + (0.203 − 0.979i)14-s + (0.962 + 0.269i)15-s + (0.682 − 0.730i)16-s + (0.962 − 0.269i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.168 + 0.985i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (300, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.168 + 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.158150907 + 0.9770247485i\)
\(L(\frac12)\) \(\approx\) \(1.158150907 + 0.9770247485i\)
\(L(1)\) \(\approx\) \(1.009482613 + 0.6554546219i\)
\(L(1)\) \(\approx\) \(1.009482613 + 0.6554546219i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.203 + 0.979i)T \)
3 \( 1 + (0.203 + 0.979i)T \)
5 \( 1 + (0.460 - 0.887i)T \)
7 \( 1 + (-0.917 - 0.398i)T \)
11 \( 1 + (0.854 - 0.519i)T \)
13 \( 1 + (0.962 - 0.269i)T \)
17 \( 1 + (0.962 - 0.269i)T \)
19 \( 1 + (0.203 + 0.979i)T \)
29 \( 1 + (0.854 - 0.519i)T \)
31 \( 1 + (-0.0682 + 0.997i)T \)
37 \( 1 + (0.682 + 0.730i)T \)
41 \( 1 + (-0.990 + 0.136i)T \)
43 \( 1 + (-0.334 + 0.942i)T \)
47 \( 1 + (-0.0682 - 0.997i)T \)
53 \( 1 + (0.962 - 0.269i)T \)
59 \( 1 + (0.203 - 0.979i)T \)
61 \( 1 + (0.460 - 0.887i)T \)
67 \( 1 + (0.854 - 0.519i)T \)
71 \( 1 + (-0.990 - 0.136i)T \)
73 \( 1 + (0.854 + 0.519i)T \)
79 \( 1 + (-0.334 + 0.942i)T \)
83 \( 1 + (0.460 - 0.887i)T \)
89 \( 1 + (-0.775 + 0.631i)T \)
97 \( 1 + (-0.990 - 0.136i)T \)
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

−23.115520167054254478344359365161, −22.42679406682444655993894393959, −21.75198176779652984425056262452, −20.69579321212056370080127987834, −19.70373366301767241272728998994, −19.16462979253849979563947186370, −18.431025811763537613700621398352, −17.84607351512039094910963907120, −16.84187782772451952816182276579, −15.25708137397321555713210729297, −14.432287665118526664932393210992, −13.65825959031427372097913457798, −13.00973296638722309297565585703, −12.06830648043619514893074305401, −11.407505710699884573230462699634, −10.298301187297729309745859333850, −9.38903169179815387453111578955, −8.6690742422405031972122105160, −7.208936002212241490342876890421, −6.33197703459222382804851151638, −5.62766794448109032330617633615, −3.86294165269656049422852610686, −3.02585323100768837516375240725, −2.19874327450552633647380064824, −1.10336351296026361411084194863, 0.98067139977839516090036682509, 3.28644646074419202958940574690, 3.829040441887682498882913174507, 4.94997600830731210401550800486, 5.83236373686174276490331447310, 6.526749125947998277660935180444, 8.12873618738077172408980327189, 8.666188521115193117520224298178, 9.69324278141754533520336851024, 10.10709228814080919008124580912, 11.73015151338228123163987792846, 12.76568134865185202708225417408, 13.76129151437992844875058405777, 14.165804223524584533322493836512, 15.36281560837221169667221531729, 16.3360523260700631207638505252, 16.47337282321123258470011847038, 17.26955315168535204300142975002, 18.475628252731578156211975873261, 19.58767884674736135762083621758, 20.494410612948896580745529452341, 21.33999028138876915392096662003, 22.00582220204972943138896966072, 23.01616454906061951253483500463, 23.42787381242598141434881900178

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