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Properties

Label 1-608-608.307-r0-0-0
Degree $1$
Conductor $608$
Sign $0.194 + 0.980i$
Analytic cond. $2.82354$
Root an. cond. $2.82354$
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  + (−0.996 − 0.0871i)3-s + (−0.906 + 0.422i)5-s + (−0.866 − 0.5i)7-s + (0.984 + 0.173i)9-s + (−0.965 − 0.258i)11-s + (−0.0871 − 0.996i)13-s + (0.939 − 0.342i)15-s + (−0.173 − 0.984i)17-s + (0.819 + 0.573i)21-s + (0.342 + 0.939i)23-s + (0.642 − 0.766i)25-s + (−0.965 − 0.258i)27-s + (0.819 − 0.573i)29-s + (−0.5 + 0.866i)31-s + (0.939 + 0.342i)33-s + ⋯
L(s)  = 1  + (−0.996 − 0.0871i)3-s + (−0.906 + 0.422i)5-s + (−0.866 − 0.5i)7-s + (0.984 + 0.173i)9-s + (−0.965 − 0.258i)11-s + (−0.0871 − 0.996i)13-s + (0.939 − 0.342i)15-s + (−0.173 − 0.984i)17-s + (0.819 + 0.573i)21-s + (0.342 + 0.939i)23-s + (0.642 − 0.766i)25-s + (−0.965 − 0.258i)27-s + (0.819 − 0.573i)29-s + (−0.5 + 0.866i)31-s + (0.939 + 0.342i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(608\)    =    \(2^{5} \cdot 19\)
Sign: $0.194 + 0.980i$
Analytic conductor: \(2.82354\)
Root analytic conductor: \(2.82354\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{608} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 608,\ (0:\ ),\ 0.194 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2366134872 + 0.1943129326i\)
\(L(\frac12)\) \(\approx\) \(0.2366134872 + 0.1943129326i\)
\(L(1)\) \(\approx\) \(0.5040776438 + 0.01194768197i\)
\(L(1)\) \(\approx\) \(0.5040776438 + 0.01194768197i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.996 - 0.0871i)T \)
5 \( 1 + (-0.906 + 0.422i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (-0.965 - 0.258i)T \)
13 \( 1 + (-0.0871 - 0.996i)T \)
17 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (0.342 + 0.939i)T \)
29 \( 1 + (0.819 - 0.573i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (-0.642 - 0.766i)T \)
43 \( 1 + (-0.906 + 0.422i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 + (-0.422 + 0.906i)T \)
59 \( 1 + (0.819 + 0.573i)T \)
61 \( 1 + (0.906 + 0.422i)T \)
67 \( 1 + (-0.819 + 0.573i)T \)
71 \( 1 + (-0.342 + 0.939i)T \)
73 \( 1 + (0.642 + 0.766i)T \)
79 \( 1 + (-0.766 + 0.642i)T \)
83 \( 1 + (0.965 - 0.258i)T \)
89 \( 1 + (-0.642 + 0.766i)T \)
97 \( 1 + (-0.173 - 0.984i)T \)
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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

−22.927712178371763295741686993940, −22.17119653641951514568498520938, −21.36613037534990012190765017406, −20.48854561214586294365721799498, −19.3119345032837623620844077272, −18.84218178165715978347676019488, −17.93284621049533942524695989000, −16.79019986116496496615367076748, −16.26663333424742060233678578102, −15.59321837598060003678755791422, −14.79994528915140550754060182472, −13.17808501715677278082118101620, −12.606422544634202557505938013180, −11.95510166051919271738670814717, −11.018174890260125818159053119788, −10.19786646440207825762451252334, −9.163165398614851019499143804108, −8.18768720578691782848033217972, −7.03427436731488564698836761950, −6.31961852550775266800942846853, −5.192724527327814245777773246791, −4.425515112550264273780777952392, −3.39009328752864191365055827953, −1.90631180213084562127800444585, −0.24913478327138028917770782300, 0.83275461348560499062453882187, 2.7948983426407514167195018446, 3.62786142490971486877532524573, 4.84871434335607886668746295139, 5.66535282567867287488361213348, 6.90780983885119299617375756439, 7.32004268233753497852281069081, 8.41708834465681832451704379735, 9.97298807546005448102980946800, 10.45249583846926537731728042069, 11.35060974728460410850773214727, 12.14861040515649267528503584864, 13.02899819639521166412822650742, 13.74819036518114770834986685709, 15.312084165658461078288182294376, 15.76085619961452637087892906814, 16.43597202152815898566526327077, 17.47774928837593074833309780968, 18.275250164648424149267228379658, 19.01407328152100817438300659238, 19.82842012488936304053237169301, 20.73860043940211397109132941038, 21.90702374642832605465800717331, 22.5716247506297212113426683458, 23.33728719185096924448831486018

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