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Properties

Label 1-608-608.389-r0-0-0
Degree $1$
Conductor $608$
Sign $0.306 + 0.951i$
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  + (0.573 + 0.819i)3-s + (0.0871 + 0.996i)5-s + (0.866 − 0.5i)7-s + (−0.342 + 0.939i)9-s + (0.965 − 0.258i)11-s + (0.819 + 0.573i)13-s + (−0.766 + 0.642i)15-s + (0.939 − 0.342i)17-s + (0.906 + 0.422i)21-s + (−0.642 − 0.766i)23-s + (−0.984 + 0.173i)25-s + (−0.965 + 0.258i)27-s + (0.906 − 0.422i)29-s + (−0.5 − 0.866i)31-s + (0.766 + 0.642i)33-s + ⋯
L(s)  = 1  + (0.573 + 0.819i)3-s + (0.0871 + 0.996i)5-s + (0.866 − 0.5i)7-s + (−0.342 + 0.939i)9-s + (0.965 − 0.258i)11-s + (0.819 + 0.573i)13-s + (−0.766 + 0.642i)15-s + (0.939 − 0.342i)17-s + (0.906 + 0.422i)21-s + (−0.642 − 0.766i)23-s + (−0.984 + 0.173i)25-s + (−0.965 + 0.258i)27-s + (0.906 − 0.422i)29-s + (−0.5 − 0.866i)31-s + (0.766 + 0.642i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.306 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.306 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(608\)    =    \(2^{5} \cdot 19\)
Sign: $0.306 + 0.951i$
Analytic conductor: \(2.82354\)
Root analytic conductor: \(2.82354\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{608} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 608,\ (0:\ ),\ 0.306 + 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.690886199 + 1.232012071i\)
\(L(\frac12)\) \(\approx\) \(1.690886199 + 1.232012071i\)
\(L(1)\) \(\approx\) \(1.398311041 + 0.5824678744i\)
\(L(1)\) \(\approx\) \(1.398311041 + 0.5824678744i\)

Euler product

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

−23.416573506712076838619149983496, −21.83552118568005888618595684893, −21.13860958579132372580676734721, −20.242392714056620836509547810621, −19.79697519225960213105427014226, −18.73803333386252925304033148023, −17.85740412865359514742210772111, −17.35210661129057513720969306248, −16.233762142322282181206396429228, −15.22661843408791155304063010597, −14.34819718071006346393869538076, −13.69985467698368405847065790047, −12.58333288205992875764714807636, −12.16950977210879319664136168404, −11.24313033579821084445514883521, −9.784748848951925277905477312928, −8.85571990609157110538303864213, −8.29475063119706114163507170935, −7.52163825231344203648638659839, −6.20308977656769768224550005054, −5.445725068748110502136611911198, −4.2209791073603900874422472122, −3.151258727713420445049747395428, −1.64419054777513553056612984107, −1.26266547034393675722230550354, 1.50450585986479972242057820470, 2.69212805419167112415742368428, 3.76979485169983344649072759583, 4.34015674621034661902285801364, 5.686280195010876534596325507116, 6.716511341891220876219050624964, 7.789665533628513947576062901847, 8.56654071064145431486274991650, 9.63398545811877319556771068825, 10.40973656722234722564275620592, 11.20063625499907323275619932205, 11.87054917859623068752106405820, 13.660922429359080532339002987516, 14.110049624116506500155135859695, 14.672470261446347473927571461445, 15.56904769884223337317391688330, 16.58003486592017148754118803211, 17.25071803267556927729294244915, 18.46880876057541610516601681225, 19.00881897607320273385271044519, 20.139994741541354332265361068250, 20.73113584168644970100091393559, 21.6265900291350424789201058983, 22.1966052627890663961989924354, 23.11964093789795943106640759274

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