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Properties

Label 1-2608-2608.1293-r0-0-0
Degree $1$
Conductor $2608$
Sign $0.997 - 0.0713i$
Analytic cond. $12.1115$
Root an. cond. $12.1115$
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  + (0.946 + 0.323i)3-s + (−0.802 − 0.597i)5-s + (0.431 + 0.902i)7-s + (0.790 + 0.612i)9-s + (0.753 − 0.657i)11-s + (−0.957 − 0.286i)13-s + (−0.565 − 0.824i)15-s + (0.973 + 0.230i)17-s + (0.995 + 0.0968i)19-s + (0.116 + 0.993i)21-s + (−0.766 − 0.642i)23-s + (0.286 + 0.957i)25-s + (0.549 + 0.835i)27-s + (0.268 − 0.963i)29-s + (−0.993 − 0.116i)31-s + ⋯
L(s)  = 1  + (0.946 + 0.323i)3-s + (−0.802 − 0.597i)5-s + (0.431 + 0.902i)7-s + (0.790 + 0.612i)9-s + (0.753 − 0.657i)11-s + (−0.957 − 0.286i)13-s + (−0.565 − 0.824i)15-s + (0.973 + 0.230i)17-s + (0.995 + 0.0968i)19-s + (0.116 + 0.993i)21-s + (−0.766 − 0.642i)23-s + (0.286 + 0.957i)25-s + (0.549 + 0.835i)27-s + (0.268 − 0.963i)29-s + (−0.993 − 0.116i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2608 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2608 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2608\)    =    \(2^{4} \cdot 163\)
Sign: $0.997 - 0.0713i$
Analytic conductor: \(12.1115\)
Root analytic conductor: \(12.1115\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2608} (1293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2608,\ (0:\ ),\ 0.997 - 0.0713i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.362590097 - 0.08443858913i\)
\(L(\frac12)\) \(\approx\) \(2.362590097 - 0.08443858913i\)
\(L(1)\) \(\approx\) \(1.449949117 + 0.04478064872i\)
\(L(1)\) \(\approx\) \(1.449949117 + 0.04478064872i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
163 \( 1 \)
good3 \( 1 + (0.946 + 0.323i)T \)
5 \( 1 + (-0.802 - 0.597i)T \)
7 \( 1 + (0.431 + 0.902i)T \)
11 \( 1 + (0.753 - 0.657i)T \)
13 \( 1 + (-0.957 - 0.286i)T \)
17 \( 1 + (0.973 + 0.230i)T \)
19 \( 1 + (0.995 + 0.0968i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (0.268 - 0.963i)T \)
31 \( 1 + (-0.993 - 0.116i)T \)
37 \( 1 + (0.448 - 0.893i)T \)
41 \( 1 + (0.875 - 0.483i)T \)
43 \( 1 + (0.154 - 0.987i)T \)
47 \( 1 + (0.713 + 0.700i)T \)
53 \( 1 + (-0.984 - 0.173i)T \)
59 \( 1 + (-0.866 + 0.5i)T \)
61 \( 1 + (0.727 - 0.686i)T \)
67 \( 1 + (0.999 + 0.0193i)T \)
71 \( 1 + (0.360 + 0.932i)T \)
73 \( 1 + (-0.0193 - 0.999i)T \)
79 \( 1 + (0.925 + 0.378i)T \)
83 \( 1 + (-0.0774 + 0.996i)T \)
89 \( 1 + (-0.657 + 0.753i)T \)
97 \( 1 + (-0.135 - 0.990i)T \)
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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

−19.63987692192173623368598061262, −18.70162830012639940693904762468, −18.12135769534785455572617248878, −17.361177863630236261530233140715, −16.45012700342543370223672980102, −15.75223448374408567180473801051, −14.76720072201414183603137492039, −14.41555002440876199377430658811, −14.02256828994552531193106447309, −12.9576594476583170655169860119, −12.10387638765587936259223141468, −11.67694612191849452963805656129, −10.66662482768232445338958775572, −9.77135601838333235320168830717, −9.38944528607200293794480525357, −8.14882494774312083616331278616, −7.54882366650861041876695415521, −7.24160857950963677130130703461, −6.46855653128431632854471148512, −5.03163177140991350383900661124, −4.23612805105565916646130115460, −3.56464660332754614371940263511, −2.89017991680576468890481629717, −1.785159419364824382091709969414, −0.98787932375115315112221826218, 0.81972419948520146083919979315, 1.92407297329840882159169430540, 2.776503138032700401266956666056, 3.65224588518154699076816664820, 4.26636340243570047869823313397, 5.20955673921330532003530050153, 5.836115643326309437600552221136, 7.21844059570980859674616033142, 7.92444084051227728401985322811, 8.30307614508448585714225555181, 9.287762488605471154305682737680, 9.545296196127382810196972855066, 10.727960385555483736377626531859, 11.559199059671846636501548932976, 12.3573546842131209863902847730, 12.64306295441538147799846086061, 14.03538883278150495178429618726, 14.30754003490575386074873513934, 15.1156028922152414179957976115, 15.7385727807168182029571852993, 16.361952237364787582203871731889, 17.05925821419098302884872421954, 18.10329885749202760016345279324, 18.97493472337588475920851274779, 19.29928179669441073843450114534

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