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Properties

Label 1-23e2-529.165-r0-0-0
Degree $1$
Conductor $529$
Sign $0.243 - 0.969i$
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  + (0.369 + 0.929i)2-s + (−0.0929 − 0.995i)3-s + (−0.726 + 0.687i)4-s + (0.879 − 0.476i)5-s + (0.890 − 0.454i)6-s + (0.323 − 0.946i)7-s + (−0.907 − 0.421i)8-s + (−0.982 + 0.185i)9-s + (0.767 + 0.640i)10-s + (−0.999 − 0.0372i)11-s + (0.751 + 0.659i)12-s + (0.992 − 0.123i)13-s + (0.998 − 0.0496i)14-s + (−0.556 − 0.831i)15-s + (0.0558 − 0.998i)16-s + (−0.263 − 0.964i)17-s + ⋯
L(s)  = 1  + (0.369 + 0.929i)2-s + (−0.0929 − 0.995i)3-s + (−0.726 + 0.687i)4-s + (0.879 − 0.476i)5-s + (0.890 − 0.454i)6-s + (0.323 − 0.946i)7-s + (−0.907 − 0.421i)8-s + (−0.982 + 0.185i)9-s + (0.767 + 0.640i)10-s + (−0.999 − 0.0372i)11-s + (0.751 + 0.659i)12-s + (0.992 − 0.123i)13-s + (0.998 − 0.0496i)14-s + (−0.556 − 0.831i)15-s + (0.0558 − 0.998i)16-s + (−0.263 − 0.964i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.243 - 0.969i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.243 - 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.041456566 - 0.8123578793i\)
\(L(\frac12)\) \(\approx\) \(1.041456566 - 0.8123578793i\)
\(L(1)\) \(\approx\) \(1.144728292 - 0.1552312651i\)
\(L(1)\) \(\approx\) \(1.144728292 - 0.1552312651i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.369 + 0.929i)T \)
3 \( 1 + (-0.0929 - 0.995i)T \)
5 \( 1 + (0.879 - 0.476i)T \)
7 \( 1 + (0.323 - 0.946i)T \)
11 \( 1 + (-0.999 - 0.0372i)T \)
13 \( 1 + (0.992 - 0.123i)T \)
17 \( 1 + (-0.263 - 0.964i)T \)
19 \( 1 + (-0.944 - 0.329i)T \)
29 \( 1 + (0.626 + 0.779i)T \)
31 \( 1 + (-0.993 + 0.111i)T \)
37 \( 1 + (-0.996 - 0.0868i)T \)
41 \( 1 + (-0.806 + 0.591i)T \)
43 \( 1 + (0.664 - 0.747i)T \)
47 \( 1 + (-0.775 - 0.631i)T \)
53 \( 1 + (0.767 - 0.640i)T \)
59 \( 1 + (-0.191 - 0.981i)T \)
61 \( 1 + (0.481 - 0.876i)T \)
67 \( 1 + (0.105 + 0.994i)T \)
71 \( 1 + (0.606 + 0.794i)T \)
73 \( 1 + (0.626 - 0.779i)T \)
79 \( 1 + (-0.999 - 0.0124i)T \)
83 \( 1 + (-0.935 - 0.352i)T \)
89 \( 1 + (0.999 - 0.0248i)T \)
97 \( 1 + (0.975 - 0.221i)T \)
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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

−23.29064268146177993047632674915, −22.55244170528948415360378480107, −21.72147973864863263592630043661, −21.06001838505476864039986115706, −20.95533260647637197170405171551, −19.55140271680939084007556160074, −18.588130365955029673581334255196, −17.96517566214951169515576879623, −17.07849551990443916956386219798, −15.66978170514686821518123281206, −15.067305209651253947324863806, −14.269878676877274134791084206150, −13.339945010272085463649415728075, −12.45471000761523515357958916476, −11.27340562810586470752195930042, −10.65809162976792816081146880631, −10.05048241472343403561225811585, −8.97487121980327206183319018249, −8.38919602680582437509851174773, −6.15564487272308037870376374314, −5.71195036247552358590193176668, −4.738771823006666132574880013093, −3.641899007008934298433151732305, −2.62680546470687523249210751679, −1.81973089778247811552744618703, 0.60375417987017946659685339500, 2.008247451237373790656723199423, 3.352340279784660258333402208462, 4.82760875538555927224762452794, 5.47005885016481968982706432059, 6.55493381750196804828199037386, 7.16899696375225093639474984346, 8.27510858310916566832873527307, 8.818640328853811639079112979869, 10.242550539683215438773854601700, 11.294265394203764072958154090013, 12.65906279567575707356421623068, 13.19069053741090086214271660373, 13.77602765021469368311860808063, 14.45132323857098619072427972082, 15.82736790052444214529148211485, 16.60612539279912404056142965836, 17.43113605328181917479166281530, 18.0086493299142709669007232366, 18.66026132716832747302470766238, 20.14607593618565057991796639156, 20.82432292333210967649298954720, 21.72463465650002409923152737747, 22.90280810340413355040671266351, 23.51315615413290316370337350068

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