↑

Properties

Label 1-2912-2912.355-r0-0-0
Degree $1$
Conductor $2912$
Sign $0.740 + 0.672i$
Analytic cond. $13.5232$
Root an. cond. $13.5232$
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  + (0.707 + 0.707i)3-s + (0.965 + 0.258i)5-s + i·9-s + (0.707 − 0.707i)11-s + (0.5 + 0.866i)15-s + (−0.5 − 0.866i)17-s + (0.707 + 0.707i)19-s + (0.866 + 0.5i)23-s + (0.866 + 0.5i)25-s + (−0.707 + 0.707i)27-s + (0.258 − 0.965i)29-s + (0.5 − 0.866i)31-s + 33-s + (0.258 + 0.965i)37-s + (0.866 − 0.5i)41-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s + (0.965 + 0.258i)5-s + i·9-s + (0.707 − 0.707i)11-s + (0.5 + 0.866i)15-s + (−0.5 − 0.866i)17-s + (0.707 + 0.707i)19-s + (0.866 + 0.5i)23-s + (0.866 + 0.5i)25-s + (−0.707 + 0.707i)27-s + (0.258 − 0.965i)29-s + (0.5 − 0.866i)31-s + 33-s + (0.258 + 0.965i)37-s + (0.866 − 0.5i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.740 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.740 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2912\)    =    \(2^{5} \cdot 7 \cdot 13\)
Sign: $0.740 + 0.672i$
Analytic conductor: \(13.5232\)
Root analytic conductor: \(13.5232\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2912} (355, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2912,\ (0:\ ),\ 0.740 + 0.672i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.866408476 + 1.106701054i\)
\(L(\frac12)\) \(\approx\) \(2.866408476 + 1.106701054i\)
\(L(1)\) \(\approx\) \(1.679649521 + 0.4241388213i\)
\(L(1)\) \(\approx\) \(1.679649521 + 0.4241388213i\)

Euler product

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

−19.16988084375707530590890286668, −18.08511087746982545718969338931, −17.84126183629771434709696025050, −17.1828843851823322206473389216, −16.31884763124269794115069906032, −15.36997105550890227548422105894, −14.52332788305539416016897503609, −14.2585377828192040616617281409, −13.29362491915064474195286960970, −12.81065134756211148374780523350, −12.272456586073096718163234556377, −11.247586598589979173042354068172, −10.387570206361142229620095983105, −9.41697797116739062030548833771, −9.11106098032332707021495970230, −8.35023190925757009728838838844, −7.386610496795009171991611066094, −6.649188322395842328278347861724, −6.205791779818494194808158233462, −5.06501200358438326021228442558, −4.33360990697354962553316008134, −3.19425310705929312570031632855, −2.50976903319905075727853799078, −1.600908371692803645468567924542, −1.06351973162777388261646198059, 1.04423179918124702126065347283, 2.07642257542592362793265763402, 2.847617298238394658074226001787, 3.516352112317892055832600138532, 4.44662564273740382958932302178, 5.28805504530318743334505987777, 5.99112613956441745492361177466, 6.86202358108007243563484430575, 7.75048970897492014375118891536, 8.60115754349778299084636483248, 9.383304130862566574169298910824, 9.66820507526921457723965813281, 10.55412929450066111786542331823, 11.260950624272913785695704684447, 11.99961695400846377588631399141, 13.28895021758030617910486380432, 13.69037282845535830526270287294, 14.15582500881132526257636953578, 14.99634239388849087496787060036, 15.58746089390296414003961994010, 16.48402890668539868970188707383, 16.99863919341523091663818483635, 17.72231083970143816518856978259, 18.8538682046612178832360060597, 18.95225293600508206329657663330

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