↑

Properties

Label 1-148-148.27-r1-0-0
Degree $1$
Conductor $148$
Sign $-0.729 - 0.683i$
Analytic cond. $15.9048$
Root an. cond. $15.9048$
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 + (0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s + (−0.5 − 0.866i)9-s − 11-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)15-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.5 − 0.866i)21-s + 23-s + (−0.5 − 0.866i)25-s − 27-s − 29-s + 31-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s + (−0.5 − 0.866i)9-s − 11-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)15-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.5 − 0.866i)21-s + 23-s + (−0.5 − 0.866i)25-s − 27-s − 29-s + 31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.729 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.729 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(148\)    =    \(2^{2} \cdot 37\)
Sign: $-0.729 - 0.683i$
Analytic conductor: \(15.9048\)
Root analytic conductor: \(15.9048\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{148} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 148,\ (1:\ ),\ -0.729 - 0.683i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7857084347 - 1.987343556i\)
\(L(\frac12)\) \(\approx\) \(0.7857084347 - 1.987343556i\)
\(L(1)\) \(\approx\) \(1.106006853 - 0.8024782208i\)
\(L(1)\) \(\approx\) \(1.106006853 - 0.8024782208i\)

Euler product

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

−28.19200031753650316391602551296, −27.20415946442553298165995385249, −26.17584630689053650235507680784, −25.67352877539279861151933481235, −24.570359538150155082469639694048, −23.16830532118207004525494962059, −22.137924736652091856552625232791, −21.21271812166247176473864068429, −20.8312975629167095176989106852, −19.109905683820829527828157291017, −18.48828507325811800850105353183, −17.262133869427069524532837421341, −15.89829250522865358252000652199, −15.14340730842829460379174195921, −14.23747454843082967723312055715, −13.28676901380774826985866388317, −11.49554023364221480195928802168, −10.72822400732024914528342417003, −9.54503975263677241703165121953, −8.66446529915663285567048063862, −7.30201969445816004712901644389, −5.75323305452680624886900338901, −4.719565635813871245376680403663, −3.06724707404183179262559896825, −2.2160359624987475540901635364, 0.7671747664044724922208135936, 1.862190150146716203906793006206, 3.49779499049010134897061985724, 5.1027030715524073556872898234, 6.29300510252570020594457640582, 7.88875914456202360700281410044, 8.27717793394293967092030473861, 9.81642795726657213862952361024, 10.98217841065363040087934557465, 12.636090877489317293954679959243, 13.08947798187380739916783237534, 14.08935770014428487993337393803, 15.22698203073145397135178052077, 16.74880627481926489749680484500, 17.51658040579170057433572272474, 18.49863260520160309923199635543, 19.648595379319453809763491631821, 20.79499580902061705733410550666, 20.97604440098307516259763109639, 23.02764010449833664502563611255, 23.701077270656826314235807109839, 24.59522577823172088838318755593, 25.47711264835154031225631650919, 26.3198504402985220926459274307, 27.60163832820139178982791908280

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