Properties

Label 1-104-104.3-r1-0-0
Degree $1$
Conductor $104$
Sign $-0.252 - 0.967i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s − 5-s + (0.5 + 0.866i)7-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s − 21-s + (0.5 − 0.866i)23-s + 25-s + 27-s + (0.5 − 0.866i)29-s − 31-s + (−0.5 − 0.866i)33-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s − 5-s + (0.5 + 0.866i)7-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s − 21-s + (0.5 − 0.866i)23-s + 25-s + 27-s + (0.5 − 0.866i)29-s − 31-s + (−0.5 − 0.866i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(104\)    =    \(2^{3} \cdot 13\)
Sign: $-0.252 - 0.967i$
Analytic conductor: \(11.1763\)
Root analytic conductor: \(11.1763\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{104} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 104,\ (1:\ ),\ -0.252 - 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1063618709 - 0.1376971945i\)
\(L(\frac12)\) \(\approx\) \(0.1063618709 - 0.1376971945i\)
\(L(1)\) \(\approx\) \(0.5940886554 + 0.1632753463i\)
\(L(1)\) \(\approx\) \(0.5940886554 + 0.1632753463i\)

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 \( 1 - T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 - T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 + T \)
89 \( 1 + (-0.5 + 0.866i)T \)
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

−29.76912272160480054923618405036, −28.977370797293421143899313146875, −27.65109166206156463111336236075, −26.9484818413248893807801138148, −25.64766246130870855281154674904, −24.165597006987811729156304123496, −23.74619755161834181109286779765, −22.926295810406278825815474299442, −21.5538633748810307627838534318, −20.12082394744342097100259275990, −19.27323525478836914581763216062, −18.313242120109969796569205169637, −17.09461333488352122949314833217, −16.24913698009209615206643463339, −14.78471399209066712643112877622, −13.54067185358479051721915731383, −12.540492483571958258220223288489, −11.2774446497121714128940766110, −10.680873510828472431905033538473, −8.39009143360655193906214765384, −7.67913108913820139348248905774, −6.481171287991067464872883548908, −4.968454736790499123644963744174, −3.47783636632375730785943321104, −1.41468158242385770488805718284, 0.08411986133245054720994378796, 2.68486877521757785473401917745, 4.3733852460211915777991201951, 5.13450707579130745836325801173, 6.82139315423648894533401586957, 8.32493325215845953722271105448, 9.42364912718815163711893867176, 10.88427184403274438530649093259, 11.66012942230346853490518546841, 12.708383683079704550171050906225, 14.73132970085012457595790689263, 15.37795712036816480169719158816, 16.21410169830345824792633488197, 17.585905294877949674859230826701, 18.53977636868290620552973125956, 19.99674204320850640721330241432, 20.88496386042209911465839462536, 21.99611932897363211946083713411, 22.93693818255658800994487110139, 23.79988537780303966265905454619, 25.09494944675192453864232564857, 26.43599956693794714972648232950, 27.25010001348416179040472765632, 28.16347437125625367836539858567, 28.74277962505070557904353662098

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