↑

Properties

Label 1-224-224.131-r0-0-0
Degree $1$
Conductor $224$
Sign $0.0612 + 0.998i$
Analytic cond. $1.04025$
Root an. cond. $1.04025$
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  + (−0.965 + 0.258i)3-s + (−0.965 − 0.258i)5-s + (0.866 − 0.5i)9-s + (−0.258 − 0.965i)11-s + (0.707 + 0.707i)13-s + 15-s + (−0.5 + 0.866i)17-s + (−0.258 + 0.965i)19-s + (−0.866 + 0.5i)23-s + (0.866 + 0.5i)25-s + (−0.707 + 0.707i)27-s + (0.707 + 0.707i)29-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)33-s + (0.965 + 0.258i)37-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)3-s + (−0.965 − 0.258i)5-s + (0.866 − 0.5i)9-s + (−0.258 − 0.965i)11-s + (0.707 + 0.707i)13-s + 15-s + (−0.5 + 0.866i)17-s + (−0.258 + 0.965i)19-s + (−0.866 + 0.5i)23-s + (0.866 + 0.5i)25-s + (−0.707 + 0.707i)27-s + (0.707 + 0.707i)29-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)33-s + (0.965 + 0.258i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0612 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0612 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.0612 + 0.998i$
Analytic conductor: \(1.04025\)
Root analytic conductor: \(1.04025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 224,\ (0:\ ),\ 0.0612 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3869047408 + 0.3638855996i\)
\(L(\frac12)\) \(\approx\) \(0.3869047408 + 0.3638855996i\)
\(L(1)\) \(\approx\) \(0.6159974145 + 0.1234872899i\)
\(L(1)\) \(\approx\) \(0.6159974145 + 0.1234872899i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 + (-0.965 - 0.258i)T \)
11 \( 1 + (-0.258 - 0.965i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.258 + 0.965i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (0.707 + 0.707i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.965 + 0.258i)T \)
41 \( 1 - iT \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.258 + 0.965i)T \)
59 \( 1 + (-0.258 - 0.965i)T \)
61 \( 1 + (-0.258 + 0.965i)T \)
67 \( 1 + (0.965 - 0.258i)T \)
71 \( 1 + iT \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.707 - 0.707i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 - T \)
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

−26.3703794182235067573490844057, −25.198440106802748488706771996508, −24.1429676014483281060764567344, −23.28157547262027196591743122295, −22.755018602534147804223718554846, −21.86853343860761454924964614879, −20.4650558408402474224417754311, −19.69297954359552739013916146450, −18.34518515285526080371799957923, −18.004766107464800626980932962645, −16.722138656102530988407439119029, −15.71800549628981074393919834147, −15.15068023980811892425343986248, −13.51041857161698687723744953975, −12.57712781577122008471947767896, −11.637714057984607096667303811874, −10.904772277156074717248666989525, −9.84026751802842040231108274653, −8.225747641329302674815686656866, −7.27400661980440764487695658167, −6.37416189533686679207222455935, −4.98976625221995516519195564907, −4.08375581402747598166376033251, −2.40493167969685122656072466855, −0.490859043382098110695154980226, 1.32291913797968397723292197233, 3.55939308795689647986232300332, 4.35556513526874192797794232046, 5.67817209336504438450305857541, 6.60301081577777860440529000195, 7.96719325401412839912247190394, 8.93817352615297392963889574338, 10.45256288390782617613423184444, 11.18319624083657165089037765822, 12.04641324206030542173231881612, 12.96691678719686333982642735241, 14.32568473482160313533193629314, 15.66947648951778431222904335955, 16.17118184714952057528698368319, 17.01082453666166884112554787888, 18.269748967272262549384083595231, 19.02187881369530943631589593352, 20.12235573446820390721470094949, 21.328823377026296974643594393776, 21.94400114504503115536017596268, 23.26027174617856702643151519155, 23.64521401831387535916236275019, 24.46271294294493727312065836693, 25.95413514670836200711270977612, 26.956418672393812718154739572061

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