↑

Properties

Label 1-76e2-5776.77-r0-0-0
Degree $1$
Conductor $5776$
Sign $-0.319 - 0.947i$
Analytic cond. $26.8236$
Root an. cond. $26.8236$
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  + (−0.475 − 0.879i)3-s + (−0.324 − 0.945i)5-s + (0.986 + 0.164i)7-s + (−0.546 + 0.837i)9-s + (0.837 − 0.546i)11-s + (−0.475 − 0.879i)13-s + (−0.677 + 0.735i)15-s + (−0.986 − 0.164i)17-s + (−0.324 − 0.945i)21-s + (0.879 + 0.475i)23-s + (−0.789 + 0.614i)25-s + (0.996 + 0.0825i)27-s + (0.164 − 0.986i)29-s + (−0.0825 + 0.996i)31-s + (−0.879 − 0.475i)33-s + ⋯
L(s)  = 1  + (−0.475 − 0.879i)3-s + (−0.324 − 0.945i)5-s + (0.986 + 0.164i)7-s + (−0.546 + 0.837i)9-s + (0.837 − 0.546i)11-s + (−0.475 − 0.879i)13-s + (−0.677 + 0.735i)15-s + (−0.986 − 0.164i)17-s + (−0.324 − 0.945i)21-s + (0.879 + 0.475i)23-s + (−0.789 + 0.614i)25-s + (0.996 + 0.0825i)27-s + (0.164 − 0.986i)29-s + (−0.0825 + 0.996i)31-s + (−0.879 − 0.475i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.319 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.319 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5776\)    =    \(2^{4} \cdot 19^{2}\)
Sign: $-0.319 - 0.947i$
Analytic conductor: \(26.8236\)
Root analytic conductor: \(26.8236\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5776} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5776,\ (0:\ ),\ -0.319 - 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9909031494 - 1.379895485i\)
\(L(\frac12)\) \(\approx\) \(0.9909031494 - 1.379895485i\)
\(L(1)\) \(\approx\) \(0.9002408880 - 0.4983505027i\)
\(L(1)\) \(\approx\) \(0.9002408880 - 0.4983505027i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.475 - 0.879i)T \)
5 \( 1 + (-0.324 - 0.945i)T \)
7 \( 1 + (0.986 + 0.164i)T \)
11 \( 1 + (0.837 - 0.546i)T \)
13 \( 1 + (-0.475 - 0.879i)T \)
17 \( 1 + (-0.986 - 0.164i)T \)
23 \( 1 + (0.879 + 0.475i)T \)
29 \( 1 + (0.164 - 0.986i)T \)
31 \( 1 + (-0.0825 + 0.996i)T \)
37 \( 1 + (0.837 - 0.546i)T \)
41 \( 1 + (0.677 - 0.735i)T \)
43 \( 1 + (0.915 + 0.401i)T \)
47 \( 1 + (0.546 + 0.837i)T \)
53 \( 1 + (0.837 - 0.546i)T \)
59 \( 1 + (0.735 + 0.677i)T \)
61 \( 1 + (0.969 + 0.245i)T \)
67 \( 1 + (-0.969 + 0.245i)T \)
71 \( 1 + (-0.245 - 0.969i)T \)
73 \( 1 + (0.986 + 0.164i)T \)
79 \( 1 + (-0.401 + 0.915i)T \)
83 \( 1 + (-0.324 + 0.945i)T \)
89 \( 1 + (0.986 - 0.164i)T \)
97 \( 1 + (0.245 - 0.969i)T \)
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

−17.816869979107231789578013615306, −17.320451868978303128281381759167, −16.75789464502693546936501408489, −16.00754257838483215989865830569, −15.15572356913959621850934135312, −14.70796989173906976807143859118, −14.4697409445831552483642415356, −13.54881252592580574469781480493, −12.478525941310888728850476566283, −11.612855271156614838487734626777, −11.44963913898774405802896056396, −10.73276227088968112485129130407, −10.1693794077482271755313912730, −9.264224974685347759318388007226, −8.84377997583669040433214859942, −7.83213932713017384068407714141, −6.96516151227543432715922751195, −6.6260498356679025249588627392, −5.71006911033986143336875267564, −4.73266818256533228458837863974, −4.3160611597270070919302406651, −3.77261575723433931468137598595, −2.676091969833522197787828874197, −2.00423254784707804159897153531, −0.83093629766503547518042217266, 0.686757853322723640114491198340, 1.08938862428670852159473112105, 2.051200763505753869800851853421, 2.78006271794933827781245128494, 4.02933455921620989548624477806, 4.64012612479521398818738642482, 5.44690704827786391214923669698, 5.818996329929065028893945827750, 6.86615159329910188194893358525, 7.53325764733824284632536590785, 8.11519656048540864889931234774, 8.78125391780354467883578568263, 9.28231867930030396483365579518, 10.51715477561635505048503751890, 11.24880299155744978812726120740, 11.591048867712479573612726973813, 12.332656991748669896040878336000, 12.85531598924968486402886718378, 13.53823387945968854814833966840, 14.17914465785182509759167532251, 14.95077649603594551105240290962, 15.69249572277895479839668528260, 16.41799657442492281039387514490, 17.13003765585853835537768886075, 17.61575125737965869426803690742

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