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Properties

Label 1-2912-2912.1403-r0-0-0
Degree $1$
Conductor $2912$
Sign $-0.749 + 0.662i$
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

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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 − 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 − i·41-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 − 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 − i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2912\)    =    \(2^{5} \cdot 7 \cdot 13\)
Sign: $-0.749 + 0.662i$
Analytic conductor: \(13.5232\)
Root analytic conductor: \(13.5232\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2912} (1403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2912,\ (0:\ ),\ -0.749 + 0.662i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3997430452 + 1.055444635i\)
\(L(\frac12)\) \(\approx\) \(0.3997430452 + 1.055444635i\)
\(L(1)\) \(\approx\) \(1.036320938 + 0.3018511575i\)
\(L(1)\) \(\approx\) \(1.036320938 + 0.3018511575i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 + (-0.965 + 0.258i)T \)
11 \( 1 + (-0.258 + 0.965i)T \)
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 \( 1 + T \)
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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

−18.883363311085065232040917059991, −18.56558843262918910969349060935, −17.47546126072151294556545078211, −16.592322293773270650189966736280, −15.96929709232514275111505513821, −15.25485663561683572425532024082, −14.75226327776852750818076519925, −13.911856536674398807175515143145, −13.175965103239430342446189093844, −12.6747900152088525437139759626, −11.754334344192241101015545243248, −11.16769191317292677919954696942, −10.17864656518774367209590212289, −9.44561151881842076986548218581, −8.50241976269796446374534614365, −8.06868060728906388576906581685, −7.64782442695082844990936944900, −6.50646207324728581298489701064, −5.85505764274069986801180244606, −4.60970347890832341968454887630, −3.827130187441295334220889311099, −3.43292631189763003488263609182, −2.345030265157360731585811943865, −1.50346753639117241492265303674, −0.30260078778914522387146927815, 1.246865452576501766425795697954, 2.576399510391346017888443578183, 2.75352653219294459057602454907, 4.08472833704027899312938769498, 4.34976278079988260471090075310, 5.24204558071359191543491495544, 6.661920519255449015850012476957, 7.25059206278596411724631006484, 7.79359049475259074414308601085, 8.66116938073194933630963941785, 9.24295990769800186314812459840, 10.08290811739343108257939952414, 10.81025312704749096731436433426, 11.53214940136703367274696430795, 12.49642863775824250311974206621, 12.959836614418549336903391139310, 13.96882934281344000790873271998, 14.53693918340806534594735079962, 15.20329915480731559471919715684, 15.85471605732622989409610925122, 16.1748948753155499768565983176, 17.40592220053658635437422065510, 18.21343809875501463022669929979, 18.710443786748829070736256715469, 19.707578758399187632962602868231

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