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Properties

Label 1-23e2-529.26-r0-0-0
Degree $1$
Conductor $529$
Sign $0.188 - 0.982i$
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  + (0.586 + 0.809i)2-s + (−0.492 + 0.870i)3-s + (−0.311 + 0.950i)4-s + (0.130 − 0.991i)5-s + (−0.993 + 0.111i)6-s + (−0.896 + 0.443i)7-s + (−0.952 + 0.305i)8-s + (−0.514 − 0.857i)9-s + (0.879 − 0.476i)10-s + (0.912 + 0.409i)11-s + (−0.673 − 0.739i)12-s + (−0.935 + 0.352i)13-s + (−0.885 − 0.465i)14-s + (0.798 + 0.601i)15-s + (−0.806 − 0.591i)16-s + (−0.596 + 0.802i)17-s + ⋯
L(s)  = 1  + (0.586 + 0.809i)2-s + (−0.492 + 0.870i)3-s + (−0.311 + 0.950i)4-s + (0.130 − 0.991i)5-s + (−0.993 + 0.111i)6-s + (−0.896 + 0.443i)7-s + (−0.952 + 0.305i)8-s + (−0.514 − 0.857i)9-s + (0.879 − 0.476i)10-s + (0.912 + 0.409i)11-s + (−0.673 − 0.739i)12-s + (−0.935 + 0.352i)13-s + (−0.885 − 0.465i)14-s + (0.798 + 0.601i)15-s + (−0.806 − 0.591i)16-s + (−0.596 + 0.802i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.188 - 0.982i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.188 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01993625020 + 0.01647131900i\)
\(L(\frac12)\) \(\approx\) \(0.01993625020 + 0.01647131900i\)
\(L(1)\) \(\approx\) \(0.6088308088 + 0.4690897668i\)
\(L(1)\) \(\approx\) \(0.6088308088 + 0.4690897668i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.586 + 0.809i)T \)
3 \( 1 + (-0.492 + 0.870i)T \)
5 \( 1 + (0.130 - 0.991i)T \)
7 \( 1 + (-0.896 + 0.443i)T \)
11 \( 1 + (0.912 + 0.409i)T \)
13 \( 1 + (-0.935 + 0.352i)T \)
17 \( 1 + (-0.596 + 0.802i)T \)
19 \( 1 + (-0.791 - 0.611i)T \)
29 \( 1 + (-0.759 - 0.650i)T \)
31 \( 1 + (0.299 - 0.954i)T \)
37 \( 1 + (-0.998 + 0.0620i)T \)
41 \( 1 + (0.626 - 0.779i)T \)
43 \( 1 + (0.369 - 0.929i)T \)
47 \( 1 + (-0.917 - 0.398i)T \)
53 \( 1 + (0.879 + 0.476i)T \)
59 \( 1 + (-0.996 + 0.0868i)T \)
61 \( 1 + (-0.834 + 0.551i)T \)
67 \( 1 + (0.988 - 0.148i)T \)
71 \( 1 + (-0.449 + 0.893i)T \)
73 \( 1 + (-0.759 + 0.650i)T \)
79 \( 1 + (-0.616 + 0.787i)T \)
83 \( 1 + (-0.404 + 0.914i)T \)
89 \( 1 + (-0.239 + 0.970i)T \)
97 \( 1 + (-0.820 - 0.571i)T \)
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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

−22.72606890784341778582360454235, −22.270614517580598459186704215455, −21.43954895647131095851371067721, −19.952158754861005048118885085604, −19.51343554713931188524357149793, −18.84616217710617854842428303674, −17.99046107292250829087729155431, −17.13211073900648709099549745232, −16.02329885848933827441648098604, −14.67635625651563015255100976149, −14.12818078292899544335682908865, −13.24822737842978651250657506733, −12.492035991273251543026360849495, −11.64280201147767450302435954909, −10.83431492927667244835985119196, −10.098304371421280886738179125694, −9.05840371884124214821424049832, −7.43148091186835756855800828846, −6.57353844883191554942128702931, −6.056602950392521905989941792138, −4.775936015902032609383230497040, −3.4547787356102983761111969354, −2.68296487815999250964355483564, −1.549875123808565317070300800291, −0.01137483638992977276928198837, 2.35348373750073919859222342461, 3.92095258878480503891308384371, 4.35745485383550928266426612341, 5.430211295516259426441249596675, 6.17404537912609171771391128814, 7.05335629617115717100342308086, 8.63008551107193520325250160298, 9.16100423214119078040687329697, 9.93525364305297664338095509693, 11.49464134324538260501832873766, 12.306478661891622919053537788628, 12.86106248678539945694324585852, 13.99564196956369031586962503146, 15.253651621375052122899690824732, 15.40260229789977605515690380774, 16.673231923506506755105310370031, 16.97382557609714646602012170107, 17.63098070909093135740873782467, 19.23615693479498951542892915336, 20.13351904856020930842042479670, 21.14039915152340275527836985622, 21.86129829787942891003878303506, 22.39396989631748826263153838322, 23.19565120893835194839526738965, 24.249094555765398884417131058269

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