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Properties

Label 1-23e2-529.231-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

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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: $\chi_{529} (231, \cdot )$
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 \( 1 + (-0.990 + 0.136i)T \)
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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

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

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