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Properties

Label 1-4176-4176.347-r0-0-0
Degree $1$
Conductor $4176$
Sign $0.976 - 0.216i$
Analytic cond. $19.3932$
Root an. cond. $19.3932$
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  + (−0.866 + 0.5i)5-s + (−0.5 + 0.866i)7-s + (0.866 + 0.5i)11-s + (−0.866 + 0.5i)13-s + 17-s − i·19-s + (0.5 + 0.866i)23-s + (0.5 − 0.866i)25-s + (−0.5 − 0.866i)31-s − i·35-s − i·37-s + (0.5 + 0.866i)41-s + (−0.866 − 0.5i)43-s + (0.5 − 0.866i)47-s + (−0.5 − 0.866i)49-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)5-s + (−0.5 + 0.866i)7-s + (0.866 + 0.5i)11-s + (−0.866 + 0.5i)13-s + 17-s − i·19-s + (0.5 + 0.866i)23-s + (0.5 − 0.866i)25-s + (−0.5 − 0.866i)31-s − i·35-s − i·37-s + (0.5 + 0.866i)41-s + (−0.866 − 0.5i)43-s + (0.5 − 0.866i)47-s + (−0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4176 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4176 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4176\)    =    \(2^{4} \cdot 3^{2} \cdot 29\)
Sign: $0.976 - 0.216i$
Analytic conductor: \(19.3932\)
Root analytic conductor: \(19.3932\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4176} (347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4176,\ (0:\ ),\ 0.976 - 0.216i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.118382665 - 0.1224828221i\)
\(L(\frac12)\) \(\approx\) \(1.118382665 - 0.1224828221i\)
\(L(1)\) \(\approx\) \(0.8670075670 + 0.1211085097i\)
\(L(1)\) \(\approx\) \(0.8670075670 + 0.1211085097i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
29 \( 1 \)
good5 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + T \)
19 \( 1 - iT \)
23 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 - iT \)
59 \( 1 + (0.866 - 0.5i)T \)
61 \( 1 + (-0.866 - 0.5i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 - T \)
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

−18.73854768789861762489510077213, −17.51980905107799858014687799746, −16.85353843164339544015537562455, −16.52235697258720894023257736961, −15.92442210412932307235158077801, −14.87985819406034600979669034889, −14.47803327878717822725014347650, −13.70299088133394455304901558737, −12.78518067396837441843729004380, −12.30158866112153065830519420269, −11.76430271868459390057625202722, −10.77204663936071724431283192571, −10.25469602052083342273998633267, −9.43969792313988746497312334, −8.69131566827676192299013855363, −7.90904637012403037135677049992, −7.36828493886122597086849310701, −6.63060241699929857187709236555, −5.74733230794805539237451573190, −4.884381684898625323504918244745, −4.131055582642983380621078145144, −3.486887654416862156827215198684, −2.86414895937234216527688475613, −1.35932894420295968346235501927, −0.80444442777434986064183764369, 0.435801340676719743158569996627, 1.79163112591174735037297878617, 2.591404542938551500691154819681, 3.37821330660903472733208689244, 4.036958115684590653502811793394, 4.953418627889596410957949957924, 5.67734709285724955854464785688, 6.7316633868177557431660052903, 7.07010684453312763141881202351, 7.86399650193299458218532478696, 8.742322173456574793351148224140, 9.53139253138567638271904820649, 9.84631433134589962419781416006, 11.10705965969163711460958777166, 11.53844305561181138581823483126, 12.22019774417086974529440834483, 12.67216684040907207429505744678, 13.67689506901417675911447138090, 14.59169835128741947207027079366, 14.96865883439188789221944784459, 15.52297677180337557658973020923, 16.362888340793756488421369838765, 16.92607419681821707747951043051, 17.75737808652844775490924757876, 18.51866798998092316248939209111

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