↑

Properties

Label 1-2496-2496.149-r0-0-0
Degree $1$
Conductor $2496$
Sign $0.948 + 0.315i$
Analytic cond. $11.5913$
Root an. cond. $11.5913$
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  + (0.923 + 0.382i)5-s + (−0.258 − 0.965i)7-s + (−0.991 + 0.130i)11-s + (0.866 + 0.5i)17-s + (0.130 − 0.991i)19-s + (0.965 + 0.258i)23-s + (0.707 + 0.707i)25-s + (0.793 + 0.608i)29-s − i·31-s + (0.130 − 0.991i)35-s + (0.130 + 0.991i)37-s + (−0.258 + 0.965i)41-s + (−0.793 + 0.608i)43-s − 47-s + (−0.866 + 0.5i)49-s + ⋯
L(s)  = 1  + (0.923 + 0.382i)5-s + (−0.258 − 0.965i)7-s + (−0.991 + 0.130i)11-s + (0.866 + 0.5i)17-s + (0.130 − 0.991i)19-s + (0.965 + 0.258i)23-s + (0.707 + 0.707i)25-s + (0.793 + 0.608i)29-s − i·31-s + (0.130 − 0.991i)35-s + (0.130 + 0.991i)37-s + (−0.258 + 0.965i)41-s + (−0.793 + 0.608i)43-s − 47-s + (−0.866 + 0.5i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $0.948 + 0.315i$
Analytic conductor: \(11.5913\)
Root analytic conductor: \(11.5913\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2496} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2496,\ (0:\ ),\ 0.948 + 0.315i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.860582676 + 0.3015585056i\)
\(L(\frac12)\) \(\approx\) \(1.860582676 + 0.3015585056i\)
\(L(1)\) \(\approx\) \(1.223879067 + 0.03951801225i\)
\(L(1)\) \(\approx\) \(1.223879067 + 0.03951801225i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (0.923 + 0.382i)T \)
7 \( 1 + (-0.258 - 0.965i)T \)
11 \( 1 + (-0.991 + 0.130i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (0.130 - 0.991i)T \)
23 \( 1 + (0.965 + 0.258i)T \)
29 \( 1 + (0.793 + 0.608i)T \)
31 \( 1 - iT \)
37 \( 1 + (0.130 + 0.991i)T \)
41 \( 1 + (-0.258 + 0.965i)T \)
43 \( 1 + (-0.793 + 0.608i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.923 - 0.382i)T \)
59 \( 1 + (0.793 - 0.608i)T \)
61 \( 1 + (0.130 - 0.991i)T \)
67 \( 1 + (0.608 - 0.793i)T \)
71 \( 1 + (-0.258 - 0.965i)T \)
73 \( 1 + (0.707 + 0.707i)T \)
79 \( 1 + iT \)
83 \( 1 + (0.923 - 0.382i)T \)
89 \( 1 + (0.965 + 0.258i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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

−19.12661379917966590117659442386, −18.73357610526787099478932261507, −18.064154973694385802046014445001, −17.34750349094613903103994028680, −16.45581724113474600851331579805, −16.03897066841016762052780512139, −15.10424825352466868816548059952, −14.43545115957901565779674907268, −13.58718525760105014132814222145, −12.96586042692706556745118961182, −12.32013983981359203895569514184, −11.62793258416840936318784168323, −10.51766638143671802395619846957, −9.94796440393026328727856961570, −9.270114710858082443465372807529, −8.50218471851629316978449522936, −7.80457616168121054828085602943, −6.761444852647685674877103585055, −5.77463385499457990597642113093, −5.5015235229501077854763684964, −4.67234761671897714247923352711, −3.38308275937391382110462634093, −2.58593197359404499112439509549, −1.93066537959683876608545015434, −0.72891511855152725519255240878, 0.93732978551709107949280941669, 1.79369799932098570352306251917, 3.04341401379490988524590705159, 3.28842569419477923232357065552, 4.91034381509108485392125301711, 5.05919911625740456827934910175, 6.4223875091066526355222214704, 6.763321137312779477838798135656, 7.705316615176669336856217426305, 8.45603816128789148007625372546, 9.60243296775755909934786096095, 10.003062845622075156723763025479, 10.74242038013606925321100303821, 11.28976274622028628148690988906, 12.592658444844168995826152700506, 13.108200622262997988242007772008, 13.70080984843557776985386726420, 14.419384159892607261117867319993, 15.12910904412947523533428048338, 16.03255753719922106062074307278, 16.74057209477442734694724214192, 17.41993057212489497735006139966, 17.98264889769571637886725845143, 18.71870878435250055321305681071, 19.519505527560299223418120387093

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