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Properties

Label 1-48e2-2304.13-r0-0-0
Degree $1$
Conductor $2304$
Sign $-0.940 - 0.339i$
Analytic cond. $10.6997$
Root an. cond. $10.6997$
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  + (−0.812 + 0.582i)5-s + (−0.442 − 0.896i)7-s + (0.0327 − 0.999i)11-s + (0.412 + 0.910i)13-s + (−0.923 − 0.382i)17-s + (0.773 − 0.634i)19-s + (0.946 − 0.321i)23-s + (0.321 − 0.946i)25-s + (−0.528 − 0.849i)29-s + (−0.965 − 0.258i)31-s + (0.881 + 0.471i)35-s + (−0.634 + 0.773i)37-s + (0.321 + 0.946i)41-s + (0.683 + 0.729i)43-s + (−0.608 − 0.793i)47-s + ⋯
L(s)  = 1  + (−0.812 + 0.582i)5-s + (−0.442 − 0.896i)7-s + (0.0327 − 0.999i)11-s + (0.412 + 0.910i)13-s + (−0.923 − 0.382i)17-s + (0.773 − 0.634i)19-s + (0.946 − 0.321i)23-s + (0.321 − 0.946i)25-s + (−0.528 − 0.849i)29-s + (−0.965 − 0.258i)31-s + (0.881 + 0.471i)35-s + (−0.634 + 0.773i)37-s + (0.321 + 0.946i)41-s + (0.683 + 0.729i)43-s + (−0.608 − 0.793i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.940 - 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.940 - 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $-0.940 - 0.339i$
Analytic conductor: \(10.6997\)
Root analytic conductor: \(10.6997\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2304,\ (0:\ ),\ -0.940 - 0.339i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06979280164 - 0.3989952489i\)
\(L(\frac12)\) \(\approx\) \(0.06979280164 - 0.3989952489i\)
\(L(1)\) \(\approx\) \(0.7496912166 - 0.1045070304i\)
\(L(1)\) \(\approx\) \(0.7496912166 - 0.1045070304i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.812 + 0.582i)T \)
7 \( 1 + (-0.442 - 0.896i)T \)
11 \( 1 + (0.0327 - 0.999i)T \)
13 \( 1 + (0.412 + 0.910i)T \)
17 \( 1 + (-0.923 - 0.382i)T \)
19 \( 1 + (0.773 - 0.634i)T \)
23 \( 1 + (0.946 - 0.321i)T \)
29 \( 1 + (-0.528 - 0.849i)T \)
31 \( 1 + (-0.965 - 0.258i)T \)
37 \( 1 + (-0.634 + 0.773i)T \)
41 \( 1 + (0.321 + 0.946i)T \)
43 \( 1 + (0.683 + 0.729i)T \)
47 \( 1 + (-0.608 - 0.793i)T \)
53 \( 1 + (-0.471 - 0.881i)T \)
59 \( 1 + (0.582 + 0.812i)T \)
61 \( 1 + (-0.227 + 0.973i)T \)
67 \( 1 + (0.729 + 0.683i)T \)
71 \( 1 + (0.831 + 0.555i)T \)
73 \( 1 + (-0.555 - 0.831i)T \)
79 \( 1 + (-0.991 - 0.130i)T \)
83 \( 1 + (0.986 - 0.162i)T \)
89 \( 1 + (-0.195 - 0.980i)T \)
97 \( 1 + (-0.258 - 0.965i)T \)
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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

−20.1399582783604750179416473905, −19.230968984852418893204772434108, −18.60110715396387888935763645502, −17.789922309174717971240271899660, −17.164965020660207220241069913847, −16.0810485840541249359747685088, −15.69675393672597017385603724104, −15.13217206022478912944490868209, −14.34829821852819543071863714902, −13.11786071174097388077189652794, −12.59765737637493203659870066117, −12.24096957683390210975907835832, −11.18300789733974574984702271303, −10.61298519991230495103237424629, −9.33503657994772390283867109593, −9.09388604612813092230264013186, −8.123504953537710198979148316306, −7.42240728652039032924296664296, −6.62497348809829844431388966207, −5.44796929132243699837948793840, −5.121446844963509719776868598200, −3.90805251385411550208245613511, −3.34711169428609023194260902918, −2.23309384072804153745233399939, −1.270470062263632812233820517151, 0.14922434684194042411979672930, 1.19107726562088386388622205744, 2.603317953753796969818003340692, 3.345593067020325648403738117811, 4.04237871043343960421169916374, 4.78340702166794046911819168900, 6.01980864783145092894280873993, 6.8401670686138724656435777527, 7.20790212459402525219365424335, 8.198005866030552946632750524755, 8.9828731273499258330627586491, 9.76136013835890997228898167345, 10.7893855337134980376672440943, 11.30208695796011125415829577956, 11.692111085916043171969086042846, 13.069182843937369251159243437060, 13.4572080496711002769454466280, 14.25133678920621567170149397818, 14.9769190385123318850452959568, 15.94447255186673508270273218993, 16.27354822612006946726990968655, 17.04202338675359373796067081895, 18.01155429945908605448607835226, 18.73277267834853172080698709829, 19.3176035858158106522646305506

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