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Properties

Label 1-23e2-529.374-r0-0-0
Degree $1$
Conductor $529$
Sign $0.639 - 0.768i$
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  + (0.995 − 0.0991i)2-s + (−0.926 + 0.375i)3-s + (0.980 − 0.197i)4-s + (−0.535 + 0.844i)5-s + (−0.885 + 0.465i)6-s + (0.586 − 0.809i)7-s + (0.955 − 0.293i)8-s + (0.717 − 0.696i)9-s + (−0.449 + 0.893i)10-s + (−0.709 − 0.704i)11-s + (−0.834 + 0.551i)12-s + (−0.860 − 0.508i)13-s + (0.503 − 0.863i)14-s + (0.179 − 0.983i)15-s + (0.922 − 0.386i)16-s + (0.626 − 0.779i)17-s + ⋯
L(s)  = 1  + (0.995 − 0.0991i)2-s + (−0.926 + 0.375i)3-s + (0.980 − 0.197i)4-s + (−0.535 + 0.844i)5-s + (−0.885 + 0.465i)6-s + (0.586 − 0.809i)7-s + (0.955 − 0.293i)8-s + (0.717 − 0.696i)9-s + (−0.449 + 0.893i)10-s + (−0.709 − 0.704i)11-s + (−0.834 + 0.551i)12-s + (−0.860 − 0.508i)13-s + (0.503 − 0.863i)14-s + (0.179 − 0.983i)15-s + (0.922 − 0.386i)16-s + (0.626 − 0.779i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.639 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.639 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.639 - 0.768i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.639 - 0.768i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.511348284 - 0.7085372661i\)
\(L(\frac12)\) \(\approx\) \(1.511348284 - 0.7085372661i\)
\(L(1)\) \(\approx\) \(1.355043152 - 0.1707059836i\)
\(L(1)\) \(\approx\) \(1.355043152 - 0.1707059836i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.995 - 0.0991i)T \)
3 \( 1 + (-0.926 + 0.375i)T \)
5 \( 1 + (-0.535 + 0.844i)T \)
7 \( 1 + (0.586 - 0.809i)T \)
11 \( 1 + (-0.709 - 0.704i)T \)
13 \( 1 + (-0.860 - 0.508i)T \)
17 \( 1 + (0.626 - 0.779i)T \)
19 \( 1 + (-0.726 - 0.687i)T \)
29 \( 1 + (0.997 - 0.0744i)T \)
31 \( 1 + (0.700 + 0.713i)T \)
37 \( 1 + (0.251 - 0.967i)T \)
41 \( 1 + (-0.743 + 0.668i)T \)
43 \( 1 + (0.437 + 0.899i)T \)
47 \( 1 + (0.203 - 0.979i)T \)
53 \( 1 + (-0.449 - 0.893i)T \)
59 \( 1 + (0.783 + 0.621i)T \)
61 \( 1 + (-0.907 + 0.421i)T \)
67 \( 1 + (0.798 - 0.601i)T \)
71 \( 1 + (0.901 + 0.432i)T \)
73 \( 1 + (0.997 + 0.0744i)T \)
79 \( 1 + (-0.966 - 0.257i)T \)
83 \( 1 + (-0.287 - 0.957i)T \)
89 \( 1 + (0.867 - 0.498i)T \)
97 \( 1 + (-0.0186 + 0.999i)T \)
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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

−23.75006935263279235278043810218, −22.927719359744446545443392327405, −21.974846842284232061010960266222, −21.2455280989029371983363805194, −20.59023602010508831607258783254, −19.36828113246100521588861901947, −18.683115830357436902894707272882, −17.231775786693298519438604721643, −16.949117574207581765002287729425, −15.73850728607505844781409766173, −15.26864566017436965689036419448, −14.1760979355185655287178167321, −12.95551075055568820293541619028, −12.216793532045506284163720272845, −12.07603345420109345711322029599, −10.95061729331969479501343922847, −9.93752879533802843913815314116, −8.23998151530018415546455267227, −7.6768625956487190535400400052, −6.50800060712797646600078888405, −5.510837002750147602498407471666, −4.85933652697059820281778228671, −4.164708412074797373299845339433, −2.403363306594192826157808314302, −1.49839781401739492702831020398, 0.73865162007698914106012730496, 2.58717020683146068987853583166, 3.52692847289116597579105170119, 4.60709026007832501750744764855, 5.20800299108025028920226460004, 6.40773283569914821578623868074, 7.17316384123788203634014040877, 7.99890975030268361176140620017, 10.07410172682321747289347381016, 10.57757086394716222304974837671, 11.31504920457600230254759633663, 11.98115116495939041162641440164, 13.018703983438716747141503506287, 14.06082618924829775565837399764, 14.80962190946547323239767194340, 15.6570808175767468061636137931, 16.348033605257535106159078053697, 17.329079721172868379318255477828, 18.231108098950060145989460731993, 19.341106292359505642362384921736, 20.15429093528756943678352262570, 21.39591565701402326394136720389, 21.543626908563980513302543964616, 22.860368183182873453886159241730, 23.08047055038720904563387794593

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