Properties

Label 1-23e2-529.372-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: $\chi_{529} (372, \cdot )$
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

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

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