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Properties

Label 1-23e2-529.215-r0-0-0
Degree $1$
Conductor $529$
Sign $0.921 + 0.387i$
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  + (0.912 + 0.409i)2-s + (0.751 − 0.659i)3-s + (0.664 + 0.747i)4-s + (0.940 + 0.340i)5-s + (0.955 − 0.293i)6-s + (0.645 + 0.763i)7-s + (0.299 + 0.954i)8-s + (0.130 − 0.991i)9-s + (0.717 + 0.696i)10-s + (0.566 − 0.824i)11-s + (0.992 + 0.123i)12-s + (−0.996 − 0.0868i)13-s + (0.275 + 0.961i)14-s + (0.931 − 0.363i)15-s + (−0.117 + 0.993i)16-s + (−0.791 − 0.611i)17-s + ⋯
L(s)  = 1  + (0.912 + 0.409i)2-s + (0.751 − 0.659i)3-s + (0.664 + 0.747i)4-s + (0.940 + 0.340i)5-s + (0.955 − 0.293i)6-s + (0.645 + 0.763i)7-s + (0.299 + 0.954i)8-s + (0.130 − 0.991i)9-s + (0.717 + 0.696i)10-s + (0.566 − 0.824i)11-s + (0.992 + 0.123i)12-s + (−0.996 − 0.0868i)13-s + (0.275 + 0.961i)14-s + (0.931 − 0.363i)15-s + (−0.117 + 0.993i)16-s + (−0.791 − 0.611i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.921 + 0.387i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.921 + 0.387i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.552513841 + 0.7172372672i\)
\(L(\frac12)\) \(\approx\) \(3.552513841 + 0.7172372672i\)
\(L(1)\) \(\approx\) \(2.507619801 + 0.3687602261i\)
\(L(1)\) \(\approx\) \(2.507619801 + 0.3687602261i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.912 + 0.409i)T \)
3 \( 1 + (0.751 - 0.659i)T \)
5 \( 1 + (0.940 + 0.340i)T \)
7 \( 1 + (0.645 + 0.763i)T \)
11 \( 1 + (0.566 - 0.824i)T \)
13 \( 1 + (-0.996 - 0.0868i)T \)
17 \( 1 + (-0.791 - 0.611i)T \)
19 \( 1 + (-0.759 - 0.650i)T \)
29 \( 1 + (-0.311 + 0.950i)T \)
31 \( 1 + (-0.972 + 0.233i)T \)
37 \( 1 + (-0.635 - 0.771i)T \)
41 \( 1 + (-0.726 - 0.687i)T \)
43 \( 1 + (-0.999 + 0.0372i)T \)
47 \( 1 + (0.460 + 0.887i)T \)
53 \( 1 + (0.717 - 0.696i)T \)
59 \( 1 + (-0.287 + 0.957i)T \)
61 \( 1 + (-0.873 + 0.487i)T \)
67 \( 1 + (0.922 - 0.386i)T \)
71 \( 1 + (0.323 + 0.946i)T \)
73 \( 1 + (-0.311 - 0.950i)T \)
79 \( 1 + (0.948 - 0.317i)T \)
83 \( 1 + (-0.998 - 0.0620i)T \)
89 \( 1 + (0.798 + 0.601i)T \)
97 \( 1 + (0.890 - 0.454i)T \)
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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

−23.339831217538102389235290698325, −22.26232094530606231358772657859, −21.689242516923903434914751110707, −20.977811150843300578632778861457, −20.14098617817277947181056304736, −19.87563401124345834597736393072, −18.59396865675592188121140505591, −17.10389505558915075717221437768, −16.825102667730094773803276087366, −15.265254528209752031754598595468, −14.78095264817862638953268510215, −14.03100069581510516351092867326, −13.31729129966639133078663843187, −12.46535012996073334500021130789, −11.26518959543001934024030399459, −10.19992127920735734485124152130, −9.87440410701391398507440729751, −8.70182697672584062519866777906, −7.4398531751521239448745387707, −6.404434314097205384416636214590, −5.07561208297701223813416718019, −4.50390658067667727698983680529, −3.66452927937807460475167318165, −2.13074016284763776180629088699, −1.777799313521141978327671840067, 1.83105485291099632013065021619, 2.45340120952733019636650139449, 3.39817909525912928393485437030, 4.83377842768235027157682894653, 5.740628431692228949397037755682, 6.69505793928829057455382667095, 7.35500475444115121210104580018, 8.682081694342487321479138016210, 9.113028296281799752112280153928, 10.76648057909268821093723609478, 11.72932182563954544947712081433, 12.61677453261846110422695679069, 13.416599120464117575218814542767, 14.22292307688471799402307595404, 14.69000453384084650320796152903, 15.48480388758462324158425891983, 16.81730693793451710521847267693, 17.64372144509985650558479365210, 18.35313279861829060476234666655, 19.46543519861835367593708465340, 20.3056891417929787623691441693, 21.32822705744594778061475452136, 21.79610868086068976526558295374, 22.53095659484712780687383397589, 23.958599252473004006568003823400

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