↑

Properties

Label 1-148-148.71-r1-0-0
Degree $1$
Conductor $148$
Sign $-0.915 - 0.403i$
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  + (−0.173 + 0.984i)3-s + (0.766 + 0.642i)5-s + (−0.766 − 0.642i)7-s + (−0.939 − 0.342i)9-s + (0.5 + 0.866i)11-s + (−0.939 + 0.342i)13-s + (−0.766 + 0.642i)15-s + (−0.939 − 0.342i)17-s + (−0.173 + 0.984i)19-s + (0.766 − 0.642i)21-s + (0.5 − 0.866i)23-s + (0.173 + 0.984i)25-s + (0.5 − 0.866i)27-s + (−0.5 − 0.866i)29-s − 31-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)3-s + (0.766 + 0.642i)5-s + (−0.766 − 0.642i)7-s + (−0.939 − 0.342i)9-s + (0.5 + 0.866i)11-s + (−0.939 + 0.342i)13-s + (−0.766 + 0.642i)15-s + (−0.939 − 0.342i)17-s + (−0.173 + 0.984i)19-s + (0.766 − 0.642i)21-s + (0.5 − 0.866i)23-s + (0.173 + 0.984i)25-s + (0.5 − 0.866i)27-s + (−0.5 − 0.866i)29-s − 31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.915 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.915 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(148\)    =    \(2^{2} \cdot 37\)
Sign: $-0.915 - 0.403i$
Analytic conductor: \(15.9048\)
Root analytic conductor: \(15.9048\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{148} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 148,\ (1:\ ),\ -0.915 - 0.403i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1135347317 + 0.5391559431i\)
\(L(\frac12)\) \(\approx\) \(-0.1135347317 + 0.5391559431i\)
\(L(1)\) \(\approx\) \(0.7092468857 + 0.3796059662i\)
\(L(1)\) \(\approx\) \(0.7092468857 + 0.3796059662i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 \)
good3 \( 1 + (-0.173 + 0.984i)T \)
5 \( 1 + (0.766 + 0.642i)T \)
7 \( 1 + (-0.766 - 0.642i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.939 + 0.342i)T \)
17 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (-0.173 + 0.984i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 - T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 - T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (-0.766 + 0.642i)T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (-0.766 - 0.642i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (0.939 + 0.342i)T \)
89 \( 1 + (0.766 - 0.642i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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

−27.56493312900977471190828594178, −26.05793837891644629390378388715, −25.18686572522341717368539920708, −24.49384037684458908316079540719, −23.71392896518758786277931004190, −22.13131593300691488153845333949, −21.82821577387511483172084004345, −20.055312872823135911757273028437, −19.43705695939785105923033907765, −18.30958354328953071376736717633, −17.31616336140202431700534987125, −16.58387588733185244756184587553, −15.15165619332374097682621970601, −13.74339465219453574093601503776, −13.02284739405557911879355664460, −12.21151060121724540052534498213, −10.97484157613249535968552836891, −9.32608164434575635059214737188, −8.66990225580361356466184963299, −7.07318482327445209649045569592, −6.06799842388325749467708307542, −5.14679786670154650360224677077, −3.0028425299595682430158342339, −1.76478624332205732944436293048, −0.197768539927425677398882189265, 2.25346318234246741340168801995, 3.6527508881312213715864585175, 4.802645092229903321136538245444, 6.23985375410800331126299859267, 7.146591631239355873505403322789, 9.12416823870153969762669402061, 9.90849089335250679568492855170, 10.59856351426130297669531573855, 11.93520642496225901142664024679, 13.32443903404578887504674435898, 14.48999199029021898411696497551, 15.168876216506069727411754572449, 16.66548475287268748808965585360, 17.11055514863337222124961041308, 18.37524308521438352485959066405, 19.75162044244960885714963023890, 20.56244443373098378258750273467, 21.7392137119058932418574741161, 22.50300746948204177708885376403, 23.071112135618470993664181941028, 24.78548947174708822447333467006, 25.743635150290416011217687024340, 26.56915214794051907244312115627, 27.21858053705859621700189361767, 28.67174294058157645380627719424

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