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Properties

Label 1-23e2-529.2-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: $\chi_{529} (2, \cdot )$
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

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

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