↑

Properties

Label 1-104-104.85-r1-0-0
Degree $1$
Conductor $104$
Sign $0.852 + 0.522i$
Analytic cond. $11.1763$
Root an. cond. $11.1763$
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  + (0.5 + 0.866i)3-s − i·5-s + (0.866 + 0.5i)7-s + (−0.5 + 0.866i)9-s + (0.866 − 0.5i)11-s + (0.866 − 0.5i)15-s + (0.5 − 0.866i)17-s + (0.866 + 0.5i)19-s − i·21-s + (0.5 + 0.866i)23-s − 25-s − 27-s + (0.5 + 0.866i)29-s + i·31-s + (0.866 + 0.5i)33-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s − i·5-s + (0.866 + 0.5i)7-s + (−0.5 + 0.866i)9-s + (0.866 − 0.5i)11-s + (0.866 − 0.5i)15-s + (0.5 − 0.866i)17-s + (0.866 + 0.5i)19-s − i·21-s + (0.5 + 0.866i)23-s − 25-s − 27-s + (0.5 + 0.866i)29-s + i·31-s + (0.866 + 0.5i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(104\)    =    \(2^{3} \cdot 13\)
Sign: $0.852 + 0.522i$
Analytic conductor: \(11.1763\)
Root analytic conductor: \(11.1763\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{104} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 104,\ (1:\ ),\ 0.852 + 0.522i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.236759832 + 0.6303608456i\)
\(L(\frac12)\) \(\approx\) \(2.236759832 + 0.6303608456i\)
\(L(1)\) \(\approx\) \(1.451448337 + 0.2630307604i\)
\(L(1)\) \(\approx\) \(1.451448337 + 0.2630307604i\)

Euler product

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

−29.840527991260432700878814939222, −28.50552509163681901649276844571, −27.11831851703321455424313473485, −26.294484868064377373242141504583, −25.28298340164223551808870984550, −24.29008695815282409653399431259, −23.30952602127103440395508913693, −22.30022421283346679285197215608, −20.89881226470421031410786958929, −19.85874024832288123670646978223, −18.89751020315781006437095139064, −17.89852160143147834916107782328, −17.092620284308696210682488215236, −15.05039195505351399408765494498, −14.45480202237723108262969211145, −13.49601574324391454158943465291, −12.0540352978683447907464458771, −11.07144767401531571026704247387, −9.64398977145536247589922611510, −8.11267509427315118281857991915, −7.225842359331305264358292094003, −6.17441852771770356552731527769, −4.135311403839220114081820612347, −2.6632793737335449946449235990, −1.24929710668520858338768658409, 1.403823710172035107420116252727, 3.27879768032600805690476593917, 4.69956570447497132579077416377, 5.54865125775224883547038200117, 7.76893389984320515965093547975, 8.837624275615170074226070415229, 9.55565381347206521411690414433, 11.17932932420051416809527767566, 12.14514213739552899728502460939, 13.78123475116610037409121034176, 14.58441453304452528974186085395, 15.89427768999975436805772249132, 16.62391610987147047315230148992, 17.86999252800457208530673577899, 19.39729623414493624713669842577, 20.36074471474487561777204712580, 21.19738146771613725193311740476, 21.98248673319561638627956895823, 23.43331975809773894600018045626, 24.90157674177774736274780048022, 25.07549798123122741050085427322, 26.862072262680713907878570596919, 27.46381059689768767915772129531, 28.260111662120274586603712660872, 29.51795872470371127149901807301

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