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Properties

Label 1-463-463.18-r0-0-0
Degree $1$
Conductor $463$
Sign $0.874 - 0.485i$
Analytic cond. $2.15016$
Root an. cond. $2.15016$
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  + (−0.591 − 0.806i)2-s + (0.882 + 0.470i)3-s + (−0.301 + 0.953i)4-s + (−0.0611 − 0.998i)5-s + (−0.142 − 0.989i)6-s + (0.882 − 0.470i)7-s + (0.947 − 0.320i)8-s + (0.557 + 0.830i)9-s + (−0.768 + 0.639i)10-s + (−0.591 + 0.806i)11-s + (−0.714 + 0.699i)12-s + (−0.222 + 0.974i)13-s + (−0.900 − 0.433i)14-s + (0.415 − 0.909i)15-s + (−0.818 − 0.574i)16-s + (0.917 + 0.396i)17-s + ⋯
L(s)  = 1  + (−0.591 − 0.806i)2-s + (0.882 + 0.470i)3-s + (−0.301 + 0.953i)4-s + (−0.0611 − 0.998i)5-s + (−0.142 − 0.989i)6-s + (0.882 − 0.470i)7-s + (0.947 − 0.320i)8-s + (0.557 + 0.830i)9-s + (−0.768 + 0.639i)10-s + (−0.591 + 0.806i)11-s + (−0.714 + 0.699i)12-s + (−0.222 + 0.974i)13-s + (−0.900 − 0.433i)14-s + (0.415 − 0.909i)15-s + (−0.818 − 0.574i)16-s + (0.917 + 0.396i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 463 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 463 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(463\)
Sign: $0.874 - 0.485i$
Analytic conductor: \(2.15016\)
Root analytic conductor: \(2.15016\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{463} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 463,\ (0:\ ),\ 0.874 - 0.485i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.414746897 - 0.3664490339i\)
\(L(\frac12)\) \(\approx\) \(1.414746897 - 0.3664490339i\)
\(L(1)\) \(\approx\) \(1.117387064 - 0.2701828922i\)
\(L(1)\) \(\approx\) \(1.117387064 - 0.2701828922i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad463 \( 1 \)
good2 \( 1 + (-0.591 - 0.806i)T \)
3 \( 1 + (0.882 + 0.470i)T \)
5 \( 1 + (-0.0611 - 0.998i)T \)
7 \( 1 + (0.882 - 0.470i)T \)
11 \( 1 + (-0.591 + 0.806i)T \)
13 \( 1 + (-0.222 + 0.974i)T \)
17 \( 1 + (0.917 + 0.396i)T \)
19 \( 1 + (0.794 - 0.607i)T \)
23 \( 1 + (0.970 + 0.242i)T \)
29 \( 1 + (-0.818 + 0.574i)T \)
31 \( 1 + (0.488 - 0.872i)T \)
37 \( 1 + (-0.992 - 0.122i)T \)
41 \( 1 + (0.986 + 0.162i)T \)
43 \( 1 + (0.970 + 0.242i)T \)
47 \( 1 + (-0.979 - 0.202i)T \)
53 \( 1 + (-0.862 + 0.505i)T \)
59 \( 1 + (0.262 + 0.965i)T \)
61 \( 1 + (-0.979 - 0.202i)T \)
67 \( 1 + (0.339 - 0.940i)T \)
71 \( 1 + (0.262 - 0.965i)T \)
73 \( 1 + (0.0203 - 0.999i)T \)
79 \( 1 + (-0.714 - 0.699i)T \)
83 \( 1 + (0.917 - 0.396i)T \)
89 \( 1 + (0.917 + 0.396i)T \)
97 \( 1 + (-0.0611 + 0.998i)T \)
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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

−24.38570731247989652372454600308, −23.2917120112205140573256824673, −22.56707135408706918699227507089, −21.22513720947916488416125321734, −20.49839134986193862219618951100, −19.19493895161252844308375144576, −18.80371177776913854886471833762, −18.09228296768254418466939282998, −17.39933356709466256840496268405, −15.91162185130178041522190998065, −15.30166254987386240817581809497, −14.32513529843965181303680590877, −14.13048447016139884975379266185, −12.84078233772450507384010547936, −11.48554195910540736558855795469, −10.50808345081239898187185496570, −9.62468462746726143023434742210, −8.480393463857991229474287816608, −7.81482594548636653709229714075, −7.25771864085528314813870058995, −5.99518584150973734565851742512, −5.16108113995024593004810985640, −3.40905900251510725171396600895, −2.47306017939143171173715614919, −1.11969450986256183543968974153, 1.29728151394968255376996721729, 2.10316590651035517884182130702, 3.4338721363033316617536266406, 4.505380360340603972996887073400, 4.99534541209004556999211000593, 7.49293603181420673714655028000, 7.76571514790068139345172334331, 8.99159870201587930219589848300, 9.4436238943099078480271339188, 10.45030118218327421271115798584, 11.4040470659417912104796926059, 12.425113380411136986303104328729, 13.28926052080330481300209505971, 14.095120534596269808017362433804, 15.19350805689756324652785057842, 16.29859825901009233961220951883, 16.97685954478161129814070321777, 17.867190922113437655403610923908, 18.983521929895199909724932019979, 19.71272611456181336297872750283, 20.5935938416021000696232633438, 20.93799508534927182060582727199, 21.554779882253953805989075341435, 22.856536423999075138396056369601, 24.02747073613528171374944513511

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