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Properties

Label 1-2912-2912.2323-r0-0-0
Degree $1$
Conductor $2912$
Sign $0.894 - 0.448i$
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  + (−0.258 + 0.965i)3-s + (−0.707 + 0.707i)5-s + (−0.866 − 0.5i)9-s + (−0.965 − 0.258i)11-s + (−0.5 − 0.866i)15-s + (−0.5 + 0.866i)17-s + (−0.965 + 0.258i)19-s + (−0.866 + 0.5i)23-s − i·25-s + (0.707 − 0.707i)27-s + (−0.258 + 0.965i)29-s + 31-s + (0.5 − 0.866i)33-s + (−0.965 − 0.258i)37-s + (−0.866 + 0.5i)41-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)3-s + (−0.707 + 0.707i)5-s + (−0.866 − 0.5i)9-s + (−0.965 − 0.258i)11-s + (−0.5 − 0.866i)15-s + (−0.5 + 0.866i)17-s + (−0.965 + 0.258i)19-s + (−0.866 + 0.5i)23-s − i·25-s + (0.707 − 0.707i)27-s + (−0.258 + 0.965i)29-s + 31-s + (0.5 − 0.866i)33-s + (−0.965 − 0.258i)37-s + (−0.866 + 0.5i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2912\)    =    \(2^{5} \cdot 7 \cdot 13\)
Sign: $0.894 - 0.448i$
Analytic conductor: \(13.5232\)
Root analytic conductor: \(13.5232\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2912} (2323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2912,\ (0:\ ),\ 0.894 - 0.448i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1768278504 - 0.04182863731i\)
\(L(\frac12)\) \(\approx\) \(0.1768278504 - 0.04182863731i\)
\(L(1)\) \(\approx\) \(0.5309242485 + 0.2849217352i\)
\(L(1)\) \(\approx\) \(0.5309242485 + 0.2849217352i\)

Euler product

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

−18.92621704568707830894654571717, −18.76843266544627587388316365266, −17.59817132426617136744403370605, −17.347958330367829527493671684971, −16.33685875688131890817714731733, −15.77148684362627098376703861272, −15.09945221232806415898084065931, −14.035634940288733771874614300495, −13.352957489515868677857139338637, −12.84168887941456349194259477810, −12.07878063235524664463442204276, −11.62064255490503541575796773339, −10.78104290113273969499500517331, −9.95390377994094141222780587657, −8.82447236451523232234064891928, −8.26947990736643669582728294937, −7.68940786436786790017885105195, −6.89650347751371013742802733802, −6.177767517255589927413170975141, −5.12502211319491632089075428055, −4.70375982769881343931597820253, −3.60823835226162918894434120965, −2.49130898275845389816929561365, −1.89607238470634693386326824313, −0.61920248597036843342824524314, 0.08988997293786658654024103993, 1.8281491231051023747478383821, 2.92753233954105988027976233563, 3.4892308844321305131315288801, 4.325890611001452200906843825678, 4.961987735239719706451120692217, 6.042063049352209632188959181836, 6.4848580164360944409657310658, 7.68288467789341650351383279558, 8.25985097054912028522999437573, 8.9938081484280441854854008054, 10.094094396325399437921739154, 10.5490128411273326856591305866, 11.03685861356837979815239283001, 11.85292867550938701821541796608, 12.558289519327057567302082432654, 13.54261550881298867795391460395, 14.35918915984707616535230499695, 15.12544346568817637495460542884, 15.4477113367676589149797431950, 16.19430278280978441092661724830, 16.80859502347786880728924749279, 17.77486271850921145497398128146, 18.22192233097788698814919701960, 19.281889597509015443494019608464

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