Properties

Label 1-2960-2960.13-r0-0-0
Degree $1$
Conductor $2960$
Sign $0.923 + 0.384i$
Analytic cond. $13.7461$
Root an. cond. $13.7461$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.939 + 0.342i)3-s + (0.984 + 0.173i)7-s + (0.766 − 0.642i)9-s + (0.866 − 0.5i)11-s + (0.642 − 0.766i)13-s + (0.766 − 0.642i)17-s + (−0.939 + 0.342i)19-s + (−0.984 + 0.173i)21-s + (−0.5 + 0.866i)23-s + (−0.5 + 0.866i)27-s + (0.5 + 0.866i)29-s i·31-s + (−0.642 + 0.766i)33-s + (−0.342 + 0.939i)39-s + (0.766 + 0.642i)41-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)3-s + (0.984 + 0.173i)7-s + (0.766 − 0.642i)9-s + (0.866 − 0.5i)11-s + (0.642 − 0.766i)13-s + (0.766 − 0.642i)17-s + (−0.939 + 0.342i)19-s + (−0.984 + 0.173i)21-s + (−0.5 + 0.866i)23-s + (−0.5 + 0.866i)27-s + (0.5 + 0.866i)29-s i·31-s + (−0.642 + 0.766i)33-s + (−0.342 + 0.939i)39-s + (0.766 + 0.642i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $0.923 + 0.384i$
Analytic conductor: \(13.7461\)
Root analytic conductor: \(13.7461\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2960,\ (0:\ ),\ 0.923 + 0.384i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.583768689 + 0.3167053835i\)
\(L(\frac12)\) \(\approx\) \(1.583768689 + 0.3167053835i\)
\(L(1)\) \(\approx\) \(1.026691981 + 0.09122490450i\)
\(L(1)\) \(\approx\) \(1.026691981 + 0.09122490450i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good3 \( 1 + (-0.939 + 0.342i)T \)
7 \( 1 + (0.984 + 0.173i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (0.642 - 0.766i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 - iT \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (0.173 + 0.984i)T \)
59 \( 1 + (-0.173 - 0.984i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (-0.173 + 0.984i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.984 - 0.173i)T \)
83 \( 1 + (-0.766 + 0.642i)T \)
89 \( 1 + (0.984 - 0.173i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.93940249359609829181463532649, −18.28750876643130146607402815545, −17.47756254612490380835255800177, −17.14439042074865800947333671046, −16.46150830711940325100961796364, −15.63784516913069039026761163533, −14.75867541765017659022013892797, −14.13856539247846031671271009396, −13.441244589075675867443655829108, −12.32981540388258097946854992945, −12.11385106075539180856005093684, −11.25039423374221384334675182615, −10.64917521999346174701351848211, −10.02009788381938483523831676348, −8.85896419932823452909480134401, −8.2701571716396555341048885976, −7.3370321590940357020827549170, −6.64597706370629274130612527935, −6.045523946821797961421242333, −5.14703505972749525230753718152, −4.300967929756532715569148872505, −3.911056931848002410231753163188, −2.20890196411609874634008641214, −1.656087225867165489076210849842, −0.763820564534871473797474345187, 0.92779518224398735620891049009, 1.460044768619122274734058546431, 2.80610080950453896509198996923, 3.86745238366115062343878853679, 4.393465493520117131748291339957, 5.462059293525946883572597048969, 5.80223353124315226225898144805, 6.647916022960036657946279798101, 7.640270288066224581956231720007, 8.30603201241628229927208936717, 9.20928086045996020669195560566, 9.9384468104978932456743186582, 10.8503308010114194279365332846, 11.242346232773207484070743713868, 11.94381713547925753695972323216, 12.57727352455042908697961459939, 13.50093502634264370341750353661, 14.403027052760141287337915861674, 14.89473530095690273350874069459, 15.81361134871303093794509257521, 16.323231250074748671572928836875, 17.21333222696008650118761185042, 17.55322629210513172469942056012, 18.39242334739435535322170435878, 18.8664651425229276936424070986

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