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Properties

Label 1-23e2-529.265-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.950 - 0.311i$
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  + (0.948 − 0.317i)2-s + (−0.449 − 0.893i)3-s + (0.798 − 0.601i)4-s + (−0.791 − 0.611i)5-s + (−0.709 − 0.704i)6-s + (0.997 + 0.0744i)7-s + (0.566 − 0.824i)8-s + (−0.596 + 0.802i)9-s + (−0.944 − 0.329i)10-s + (−0.982 − 0.185i)11-s + (−0.896 − 0.443i)12-s + (0.813 − 0.581i)13-s + (0.969 − 0.245i)14-s + (−0.191 + 0.981i)15-s + (0.275 − 0.961i)16-s + (−0.972 − 0.233i)17-s + ⋯
L(s)  = 1  + (0.948 − 0.317i)2-s + (−0.449 − 0.893i)3-s + (0.798 − 0.601i)4-s + (−0.791 − 0.611i)5-s + (−0.709 − 0.704i)6-s + (0.997 + 0.0744i)7-s + (0.566 − 0.824i)8-s + (−0.596 + 0.802i)9-s + (−0.944 − 0.329i)10-s + (−0.982 − 0.185i)11-s + (−0.896 − 0.443i)12-s + (0.813 − 0.581i)13-s + (0.969 − 0.245i)14-s + (−0.191 + 0.981i)15-s + (0.275 − 0.961i)16-s + (−0.972 − 0.233i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.950 - 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.950 - 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.950 - 0.311i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (265, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.950 - 0.311i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2695349346 - 1.686744022i\)
\(L(\frac12)\) \(\approx\) \(0.2695349346 - 1.686744022i\)
\(L(1)\) \(\approx\) \(1.027936300 - 0.9738939625i\)
\(L(1)\) \(\approx\) \(1.027936300 - 0.9738939625i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.948 - 0.317i)T \)
3 \( 1 + (-0.449 - 0.893i)T \)
5 \( 1 + (-0.791 - 0.611i)T \)
7 \( 1 + (0.997 + 0.0744i)T \)
11 \( 1 + (-0.982 - 0.185i)T \)
13 \( 1 + (0.813 - 0.581i)T \)
17 \( 1 + (-0.972 - 0.233i)T \)
19 \( 1 + (0.105 - 0.994i)T \)
29 \( 1 + (-0.239 - 0.970i)T \)
31 \( 1 + (-0.847 + 0.530i)T \)
37 \( 1 + (-0.907 - 0.421i)T \)
41 \( 1 + (0.999 - 0.0248i)T \)
43 \( 1 + (-0.471 + 0.882i)T \)
47 \( 1 + (0.962 + 0.269i)T \)
53 \( 1 + (-0.944 + 0.329i)T \)
59 \( 1 + (-0.820 - 0.571i)T \)
61 \( 1 + (0.586 + 0.809i)T \)
67 \( 1 + (0.503 + 0.863i)T \)
71 \( 1 + (-0.117 - 0.993i)T \)
73 \( 1 + (-0.239 + 0.970i)T \)
79 \( 1 + (-0.998 - 0.0620i)T \)
83 \( 1 + (0.227 - 0.973i)T \)
89 \( 1 + (0.992 - 0.123i)T \)
97 \( 1 + (0.437 - 0.899i)T \)
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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

−23.63345972447352484689945329554, −23.10857808651183540304276706113, −22.21027899731157131513608805612, −21.52933254013256756430215579628, −20.63673130670421561684482158007, −20.23476074578175737157081523485, −18.66897078699961891321810103487, −17.82774472200308764690595936045, −16.78351182664182911312846422847, −15.91368242832780898505009195676, −15.38391989705787868343742745098, −14.61901479715744075217482334129, −13.8914698895491989480016223821, −12.583411072327608370988791260675, −11.66470132003895629150361972663, −10.954547942901082657696237367295, −10.53586391740706709841889162673, −8.76746227017882754796081864909, −7.89509316240065463223121483992, −6.91264563274581033621595831865, −5.84694660837875778479884353140, −4.924959081886643306850923413071, −4.10676574142810414791377804850, −3.39684301831190690569726558349, −2.02441488883629874971646770346, 0.68486475730205795429764166951, 1.8527651152586567156683706200, 2.941114545147305054827111062446, 4.36495423408387626831670677285, 5.11989503429944992835390756174, 5.88976915387770237567638147587, 7.18562739090824771454358594042, 7.86824430814926125590896376539, 8.83361277676681148940178891666, 10.781893566944644769504491212603, 11.144294568416084389269857806227, 11.91631821185312295313626089181, 12.94273560243924478477835711400, 13.30357531202427102519164642429, 14.35465622023453859348554080571, 15.583556272231067780845639357512, 15.92978510672285344422209396873, 17.28073977733375524775767349233, 18.1338239403217328719679610479, 19.00344979740767904287168013438, 19.95953003904435339702116006612, 20.54686593816898516243993819225, 21.42243831537747717919143428793, 22.485364310873383186900010182013, 23.31961384112567026853086542023

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