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Properties

Label 1-23e2-529.196-r0-0-0
Degree $1$
Conductor $529$
Sign $0.999 + 0.0118i$
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  + (0.879 + 0.476i)2-s + (0.346 − 0.938i)3-s + (0.545 + 0.837i)4-s + (−0.311 + 0.950i)5-s + (0.751 − 0.659i)6-s + (0.00620 − 0.999i)7-s + (0.0806 + 0.996i)8-s + (−0.759 − 0.650i)9-s + (−0.726 + 0.687i)10-s + (0.717 − 0.696i)11-s + (0.975 − 0.221i)12-s + (0.890 + 0.454i)13-s + (0.481 − 0.876i)14-s + (0.783 + 0.621i)15-s + (−0.404 + 0.914i)16-s + (−0.239 − 0.970i)17-s + ⋯
L(s)  = 1  + (0.879 + 0.476i)2-s + (0.346 − 0.938i)3-s + (0.545 + 0.837i)4-s + (−0.311 + 0.950i)5-s + (0.751 − 0.659i)6-s + (0.00620 − 0.999i)7-s + (0.0806 + 0.996i)8-s + (−0.759 − 0.650i)9-s + (−0.726 + 0.687i)10-s + (0.717 − 0.696i)11-s + (0.975 − 0.221i)12-s + (0.890 + 0.454i)13-s + (0.481 − 0.876i)14-s + (0.783 + 0.621i)15-s + (−0.404 + 0.914i)16-s + (−0.239 − 0.970i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.999 + 0.0118i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (196, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.999 + 0.0118i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.597762337 + 0.01542761251i\)
\(L(\frac12)\) \(\approx\) \(2.597762337 + 0.01542761251i\)
\(L(1)\) \(\approx\) \(1.908613908 + 0.07612902124i\)
\(L(1)\) \(\approx\) \(1.908613908 + 0.07612902124i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.879 + 0.476i)T \)
3 \( 1 + (0.346 - 0.938i)T \)
5 \( 1 + (-0.311 + 0.950i)T \)
7 \( 1 + (0.00620 - 0.999i)T \)
11 \( 1 + (0.717 - 0.696i)T \)
13 \( 1 + (0.890 + 0.454i)T \)
17 \( 1 + (-0.239 - 0.970i)T \)
19 \( 1 + (0.798 - 0.601i)T \)
29 \( 1 + (0.931 + 0.363i)T \)
31 \( 1 + (-0.673 + 0.739i)T \)
37 \( 1 + (0.955 + 0.293i)T \)
41 \( 1 + (0.867 + 0.498i)T \)
43 \( 1 + (0.767 + 0.640i)T \)
47 \( 1 + (0.854 + 0.519i)T \)
53 \( 1 + (-0.726 - 0.687i)T \)
59 \( 1 + (-0.977 + 0.209i)T \)
61 \( 1 + (0.566 - 0.824i)T \)
67 \( 1 + (-0.996 - 0.0868i)T \)
71 \( 1 + (-0.616 - 0.787i)T \)
73 \( 1 + (0.931 - 0.363i)T \)
79 \( 1 + (-0.263 + 0.964i)T \)
83 \( 1 + (-0.993 + 0.111i)T \)
89 \( 1 + (-0.860 + 0.508i)T \)
97 \( 1 + (-0.0929 - 0.995i)T \)
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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

−23.24429704067247648120232201534, −22.48563701958133024474353153214, −21.74533772721035757091795218538, −20.98080325871524921108101936496, −20.31683744765418950146121411378, −19.728666287272534385806298020435, −18.77303982370806745895324520873, −17.427112083631365469059119414763, −16.27353863818642073162462384295, −15.63991675180711993026090581432, −15.03466557121468078222684952361, −14.15952240875333826332620087054, −13.083977959245251925574972773361, −12.27895678782065392647781113403, −11.558695041661285175681534121119, −10.56065916185474225511015462428, −9.49975046195902414714990904044, −8.88315199394453096863055296701, −7.79407336646568124982674383018, −6.00698249508253220028598705489, −5.4846178278439493999554291926, −4.31793450645927686468010101631, −3.82992913991310802040388386196, −2.583121592247419908087113197424, −1.42413878272654583533818815119, 1.20955579783382358006248180776, 2.79060049686473035820375090493, 3.407721334595104512056144290507, 4.424535339690384222180203937595, 5.99955511968809125263182240779, 6.717053844651967129054661450227, 7.28149886780160687727281621217, 8.13183666651463639802164484130, 9.27418834816320546389040739449, 11.13016404856055627086524363537, 11.29469282047971733243401181444, 12.45721725147762463032515207425, 13.63750890882790835290706972963, 13.94717957391729776420642629772, 14.54143868777668224757997553506, 15.79059275326826126118908468373, 16.50805779131853836546280657098, 17.68012772839180903821019257160, 18.280415996944101609927931816421, 19.49014109876565178869678660436, 20.02532549182356691286844380918, 21.01496923953694434324260769420, 22.11896306669664288744212420040, 22.84213631266675113761342862980, 23.55044306419240548482585755679

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