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Properties

Label 1-2404-2404.1151-r0-0-0
Degree $1$
Conductor $2404$
Sign $0.496 - 0.868i$
Analytic cond. $11.1641$
Root an. cond. $11.1641$
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  + (−0.0627 + 0.998i)3-s + i·5-s + (−0.397 + 0.917i)7-s + (−0.992 − 0.125i)9-s + (−0.960 + 0.278i)11-s + (0.951 + 0.309i)13-s + (−0.998 − 0.0627i)15-s + (−0.156 + 0.987i)17-s + (0.860 − 0.509i)19-s + (−0.891 − 0.453i)21-s + (−0.684 − 0.728i)23-s − 25-s + (0.187 − 0.982i)27-s + (−0.509 − 0.860i)29-s + (−0.999 − 0.0314i)31-s + ⋯
L(s)  = 1  + (−0.0627 + 0.998i)3-s + i·5-s + (−0.397 + 0.917i)7-s + (−0.992 − 0.125i)9-s + (−0.960 + 0.278i)11-s + (0.951 + 0.309i)13-s + (−0.998 − 0.0627i)15-s + (−0.156 + 0.987i)17-s + (0.860 − 0.509i)19-s + (−0.891 − 0.453i)21-s + (−0.684 − 0.728i)23-s − 25-s + (0.187 − 0.982i)27-s + (−0.509 − 0.860i)29-s + (−0.999 − 0.0314i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2404 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.496 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2404 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.496 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2404\)    =    \(2^{2} \cdot 601\)
Sign: $0.496 - 0.868i$
Analytic conductor: \(11.1641\)
Root analytic conductor: \(11.1641\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2404} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2404,\ (0:\ ),\ 0.496 - 0.868i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001904031119 + 0.001104850934i\)
\(L(\frac12)\) \(\approx\) \(0.001904031119 + 0.001104850934i\)
\(L(1)\) \(\approx\) \(0.6212952425 + 0.4402654083i\)
\(L(1)\) \(\approx\) \(0.6212952425 + 0.4402654083i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
601 \( 1 \)
good3 \( 1 + (-0.0627 + 0.998i)T \)
5 \( 1 + iT \)
7 \( 1 + (-0.397 + 0.917i)T \)
11 \( 1 + (-0.960 + 0.278i)T \)
13 \( 1 + (0.951 + 0.309i)T \)
17 \( 1 + (-0.156 + 0.987i)T \)
19 \( 1 + (0.860 - 0.509i)T \)
23 \( 1 + (-0.684 - 0.728i)T \)
29 \( 1 + (-0.509 - 0.860i)T \)
31 \( 1 + (-0.999 - 0.0314i)T \)
37 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (0.750 - 0.661i)T \)
43 \( 1 + (-0.790 - 0.612i)T \)
47 \( 1 + (0.187 - 0.982i)T \)
53 \( 1 + (0.612 + 0.790i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (0.368 + 0.929i)T \)
67 \( 1 + (0.929 + 0.368i)T \)
71 \( 1 + (-0.987 + 0.156i)T \)
73 \( 1 + (-0.975 - 0.218i)T \)
79 \( 1 + (-0.0941 + 0.995i)T \)
83 \( 1 + (-0.827 - 0.562i)T \)
89 \( 1 + (0.187 + 0.982i)T \)
97 \( 1 + (0.891 + 0.453i)T \)
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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

−18.86285840605347280445671148878, −18.14534108215761097513879302126, −17.73399500322500123302975868424, −16.658674560140098254808742918273, −16.27473759633156999560365341563, −15.65492722950857215710312892420, −14.28236574010083764059941166120, −13.69036587399815760352062045339, −13.11637419011113813788261096050, −12.745183369650565323728286638833, −11.69434961812896718489131318308, −11.16254897444316328891437637749, −10.16503827685431720889773303126, −9.34109937368149149566761853046, −8.49196733827973512727531404075, −7.732209625533409895748181106462, −7.33489994918524912785466569058, −6.21432623579175857010288891194, −5.55260570782188501394164291182, −4.8323657427672208561038870075, −3.6179386141765674024372288706, −2.97621541810050754313074720956, −1.64088630976083639517558107529, −1.05031178772073811649198679186, −0.00076407789706564573825332915, 2.06129243710203809138982902022, 2.651070322716105710205477764, 3.58521219746980061002164336534, 4.12645806600957881199500436339, 5.460602438847903931003404148345, 5.78057903145680005100960920662, 6.65900451754088888860946612476, 7.66870846583344608817831433942, 8.57997324268335001930002032417, 9.20564658343539402149847812513, 10.09950006269526467107701558241, 10.58941015469919566093441797600, 11.29575840209283366815410851197, 11.99137144139680749101729726999, 12.98434309911116868665935194273, 13.79199147045991705771004748009, 14.618417650057028156956037672042, 15.2549063630349093883835519279, 15.75435673819048642068376404882, 16.2678778143890901497713864686, 17.33297481294557847370947488464, 18.15180757209525730841632620597, 18.55360610009086041888884074319, 19.386832753193292006725642493959, 20.24077081661777377518391169992

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