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Properties

Label 1-104-104.83-r0-0-0
Degree $1$
Conductor $104$
Sign $0.957 + 0.289i$
Analytic cond. $0.482973$
Root an. cond. $0.482973$
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  + 3-s + i·5-s − i·7-s + 9-s + i·11-s + i·15-s − 17-s − i·19-s − i·21-s + 23-s − 25-s + 27-s − 29-s + i·31-s + i·33-s + ⋯
L(s)  = 1  + 3-s + i·5-s − i·7-s + 9-s + i·11-s + i·15-s − 17-s − i·19-s − i·21-s + 23-s − 25-s + 27-s − 29-s + i·31-s + i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(104\)    =    \(2^{3} \cdot 13\)
Sign: $0.957 + 0.289i$
Analytic conductor: \(0.482973\)
Root analytic conductor: \(0.482973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{104} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 104,\ (0:\ ),\ 0.957 + 0.289i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.353285148 + 0.2003792255i\)
\(L(\frac12)\) \(\approx\) \(1.353285148 + 0.2003792255i\)
\(L(1)\) \(\approx\) \(1.340819274 + 0.1222351394i\)
\(L(1)\) \(\approx\) \(1.340819274 + 0.1222351394i\)

Euler product

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

−29.674232190497932432114579967166, −28.69402813427891204391527536587, −27.55822657227158290083426828898, −26.67190525776463214437247193299, −25.35286070060233480586330733496, −24.68226640062833262804760906426, −23.95020965508471462551933742236, −22.16937967849935775834927307622, −21.18260578555360946390413246798, −20.40635196764084746049844438977, −19.236253314494208800094904726940, −18.451979910390451737833758465724, −16.83172817169397830478634521179, −15.77823303970375977058538857226, −14.842076631558094647979424002838, −13.4841632452181594313968882513, −12.72220437874377875621243568453, −11.38377170768371027752287017116, −9.58273439530225293915985266731, −8.76267925528309624464523234034, −7.965323736850500872950846808986, −6.10154310517982721280906856804, −4.69355359549086374781065187417, −3.20507677988136121009162661556, −1.724412151392033829293461494619, 2.03072085902035523894960830311, 3.35489772875109672575053054801, 4.559555184206404695147635797873, 6.91485074870889498954188062482, 7.335227418704882966370140935417, 8.96917838912562181865883221148, 10.13852694553232844506891424545, 11.08905611639279114456273065815, 12.909812046417748143657689521475, 13.82652782389433190536361593907, 14.82809984237533093608154812269, 15.62520015016046470756969423129, 17.31851802470837248186850658675, 18.30990733185041872671535930486, 19.55609135108431760248377533269, 20.18554100362887034844933013001, 21.39185541716546912804053875883, 22.56968353268911529084545265877, 23.57769009888646950811035770708, 24.83685905199266756095789592587, 25.970799602580973314276510595703, 26.47160104812835863038380621998, 27.41222370155879103930673000574, 28.96868823316552526010648448425, 30.251409495054126563303729895257

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