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Properties

Label 1-1216-1216.693-r0-0-0
Degree $1$
Conductor $1216$
Sign $0.407 - 0.913i$
Analytic cond. $5.64708$
Root an. cond. $5.64708$
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  + (−0.216 − 0.976i)3-s + (−0.887 + 0.461i)5-s + (0.258 − 0.965i)7-s + (−0.906 + 0.422i)9-s + (0.793 − 0.608i)11-s + (0.843 − 0.537i)13-s + (0.642 + 0.766i)15-s + (0.342 + 0.939i)17-s + (−0.999 − 0.0436i)21-s + (−0.0871 + 0.996i)23-s + (0.573 − 0.819i)25-s + (0.608 + 0.793i)27-s + (0.999 − 0.0436i)29-s + (0.5 + 0.866i)31-s + (−0.766 − 0.642i)33-s + ⋯
L(s)  = 1  + (−0.216 − 0.976i)3-s + (−0.887 + 0.461i)5-s + (0.258 − 0.965i)7-s + (−0.906 + 0.422i)9-s + (0.793 − 0.608i)11-s + (0.843 − 0.537i)13-s + (0.642 + 0.766i)15-s + (0.342 + 0.939i)17-s + (−0.999 − 0.0436i)21-s + (−0.0871 + 0.996i)23-s + (0.573 − 0.819i)25-s + (0.608 + 0.793i)27-s + (0.999 − 0.0436i)29-s + (0.5 + 0.866i)31-s + (−0.766 − 0.642i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.407 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.407 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.407 - 0.913i$
Analytic conductor: \(5.64708\)
Root analytic conductor: \(5.64708\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (693, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1216,\ (0:\ ),\ 0.407 - 0.913i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.103915964 - 0.7164724392i\)
\(L(\frac12)\) \(\approx\) \(1.103915964 - 0.7164724392i\)
\(L(1)\) \(\approx\) \(0.9215327119 - 0.3298206097i\)
\(L(1)\) \(\approx\) \(0.9215327119 - 0.3298206097i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.216 - 0.976i)T \)
5 \( 1 + (-0.887 + 0.461i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
11 \( 1 + (0.793 - 0.608i)T \)
13 \( 1 + (0.843 - 0.537i)T \)
17 \( 1 + (0.342 + 0.939i)T \)
23 \( 1 + (-0.0871 + 0.996i)T \)
29 \( 1 + (0.999 - 0.0436i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.382 - 0.923i)T \)
41 \( 1 + (0.573 + 0.819i)T \)
43 \( 1 + (0.461 + 0.887i)T \)
47 \( 1 + (-0.342 + 0.939i)T \)
53 \( 1 + (-0.953 + 0.300i)T \)
59 \( 1 + (0.0436 - 0.999i)T \)
61 \( 1 + (0.461 - 0.887i)T \)
67 \( 1 + (-0.999 + 0.0436i)T \)
71 \( 1 + (0.0871 + 0.996i)T \)
73 \( 1 + (-0.573 - 0.819i)T \)
79 \( 1 + (0.984 + 0.173i)T \)
83 \( 1 + (0.608 - 0.793i)T \)
89 \( 1 + (0.573 - 0.819i)T \)
97 \( 1 + (0.939 - 0.342i)T \)
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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

−20.99586597183255633207697122020, −20.79130727272635966405771557367, −19.9022366050844647731814038143, −19.031885562214513395399521833249, −18.25158900949205383043763063929, −17.313342710099192394712019259279, −16.425754614095513458906503282613, −15.97509924582678309852207132406, −15.18754981667444808437930317309, −14.63878304495084363709960443541, −13.67581513604856097997560670818, −12.293644193964150513365960830613, −11.88379265516984388460935466018, −11.31332832268479118673882591487, −10.27742071861014942256906167455, −9.25557726637716642051565951519, −8.81654739859623842955690200125, −8.02554045568714522494068967252, −6.77260639257848106018538985931, −5.88932872686942604439357728595, −4.84674313505772751070656944433, −4.356249100044355776420746238059, −3.45084957894292842534324662717, −2.37836751118584390770927618248, −0.90046517989491818821344564280, 0.81363908303927047796536607801, 1.48923785584905241022514691548, 3.06767549487814277588429808671, 3.66174188501193976725139270768, 4.70365826999818007666515181656, 6.107975164577112429973735962387, 6.4863200370830520122356861876, 7.65417999762120972182023963120, 7.93963448349811367220020859589, 8.8573241052604293957645055264, 10.284996690129868863234791643375, 11.08809086798234662763534214040, 11.46285924042095682744189720642, 12.47180876379538374162037171726, 13.18016104104243098860136082702, 14.17386287296344148473863809757, 14.47249083735938555392463780560, 15.76199331121508466184128144588, 16.407515745198017208902103452669, 17.43868168303134959556638442464, 17.81261404716767441282431324152, 18.86157847561164066285264677786, 19.60735073099332037952920347606, 19.76363338321096393776894045510, 20.928350352573367964890089963623

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