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Properties

Label 1-608-608.13-r1-0-0
Degree $1$
Conductor $608$
Sign $0.980 - 0.195i$
Analytic cond. $65.3386$
Root an. cond. $65.3386$
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  + (0.0871 + 0.996i)3-s + (−0.422 + 0.906i)5-s + (−0.866 + 0.5i)7-s + (−0.984 + 0.173i)9-s + (−0.258 − 0.965i)11-s + (−0.996 − 0.0871i)13-s + (−0.939 − 0.342i)15-s + (−0.173 + 0.984i)17-s + (−0.573 − 0.819i)21-s + (0.342 − 0.939i)23-s + (−0.642 − 0.766i)25-s + (−0.258 − 0.965i)27-s + (−0.573 + 0.819i)29-s + (0.5 + 0.866i)31-s + (0.939 − 0.342i)33-s + ⋯
L(s)  = 1  + (0.0871 + 0.996i)3-s + (−0.422 + 0.906i)5-s + (−0.866 + 0.5i)7-s + (−0.984 + 0.173i)9-s + (−0.258 − 0.965i)11-s + (−0.996 − 0.0871i)13-s + (−0.939 − 0.342i)15-s + (−0.173 + 0.984i)17-s + (−0.573 − 0.819i)21-s + (0.342 − 0.939i)23-s + (−0.642 − 0.766i)25-s + (−0.258 − 0.965i)27-s + (−0.573 + 0.819i)29-s + (0.5 + 0.866i)31-s + (0.939 − 0.342i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(608\)    =    \(2^{5} \cdot 19\)
Sign: $0.980 - 0.195i$
Analytic conductor: \(65.3386\)
Root analytic conductor: \(65.3386\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{608} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 608,\ (1:\ ),\ 0.980 - 0.195i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4475777066 - 0.04423004585i\)
\(L(\frac12)\) \(\approx\) \(0.4475777066 - 0.04423004585i\)
\(L(1)\) \(\approx\) \(0.6214010862 + 0.3379847633i\)
\(L(1)\) \(\approx\) \(0.6214010862 + 0.3379847633i\)

Euler product

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

−22.983381143096964132768836986631, −22.43520252623995196895649221847, −20.88518234291351952309830233549, −20.27856925266083512297199304263, −19.47973304693454657916875124296, −19.057592893176677866305536280299, −17.71029457092529288102374316848, −17.2308320103860140150613698353, −16.28561377932796500483804059250, −15.42743833190470956135352561504, −14.34124415476807647823475477267, −13.292824079788800513493353034471, −12.83099650043539055242975481858, −12.062934448464408479119304932936, −11.2562967349813705488289234514, −9.652377956990010017958051789177, −9.31696356640113181754960742974, −7.77588704321074614011493612330, −7.48562637132333037854226264677, −6.456673727606201814974390353239, −5.271524445928506549491212916167, −4.33027176158560652199438455255, −3.03894932565020337239516729114, −1.96006826438422406435714837493, −0.685724175909480629628670050504, 0.157638223215844402259781368446, 2.479531894186618771992580387287, 3.15974769502461378523133832865, 3.97392246743991743417461008491, 5.23879811740146625928975749856, 6.13472086647243931185570358377, 7.087600549943211334767203684417, 8.376376653628644043842950849997, 9.04583895148945545170531188942, 10.32610768214042860876904259847, 10.53562363198715258619032723051, 11.685550439907646313245715812529, 12.55237107513670496071854724610, 13.78018692552740811318557656238, 14.69419579430715830633526588144, 15.25614106830896540819447607827, 16.0701975611051142489328113709, 16.7634054106437338777716487621, 17.81924047591882234244571472638, 19.154061786569759219764474597978, 19.215331850677351195061309732367, 20.3565257178847993459952536477, 21.447589144475076701779036190420, 22.11829191227231452084723987539, 22.453914913120139771852689973467

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