↑

Properties

Label 1-252-252.31-r0-0-0
Degree $1$
Conductor $252$
Sign $-0.296 + 0.954i$
Analytic cond. $1.17028$
Root an. cond. $1.17028$
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  − 5-s − 11-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s − 23-s + 25-s + (−0.5 + 0.866i)29-s + (−0.5 + 0.866i)31-s + (−0.5 + 0.866i)37-s + (0.5 + 0.866i)41-s + (0.5 − 0.866i)43-s + (−0.5 − 0.866i)47-s + (−0.5 − 0.866i)53-s + 55-s + ⋯
L(s)  = 1  − 5-s − 11-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s − 23-s + 25-s + (−0.5 + 0.866i)29-s + (−0.5 + 0.866i)31-s + (−0.5 + 0.866i)37-s + (0.5 + 0.866i)41-s + (0.5 − 0.866i)43-s + (−0.5 − 0.866i)47-s + (−0.5 − 0.866i)53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.296 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.296 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.296 + 0.954i$
Analytic conductor: \(1.17028\)
Root analytic conductor: \(1.17028\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 252,\ (0:\ ),\ -0.296 + 0.954i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3732483231 + 0.5068380428i\)
\(L(\frac12)\) \(\approx\) \(0.3732483231 + 0.5068380428i\)
\(L(1)\) \(\approx\) \(0.7317978170 + 0.1764240453i\)
\(L(1)\) \(\approx\) \(0.7317978170 + 0.1764240453i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - T \)
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 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
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 \( 1 - T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
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

−25.9281833408590200354678294270, −24.711920315658393560550025942664, −23.79371625441880237140101355203, −23.0750386140994440101429052609, −22.25752487583870533318295608892, −20.90879351911007016482323711864, −20.29350693623713164544338872864, −19.22084261768033976529268448485, −18.410834327278428077923786763650, −17.44837575173995062207110231529, −16.01833767849340253334173851616, −15.680501331816391263176536805522, −14.561240246088001123626023714787, −13.30862686291296592640197797788, −12.472510672916797711397148414567, −11.34497732808483582636230360359, −10.58467193471763012913877827592, −9.28830758350726455437527422230, −8.00214093816287286300273544809, −7.50087397188713151072006201567, −5.980957734483107114689215729961, −4.82612719416895115176687516841, −3.647395300479545914867846102792, −2.50200869500880636764024157536, −0.43738198937711023265849958962, 1.70501961564681935323603102163, 3.34030257849857660290360811973, 4.242864626316784446735920836629, 5.54926594744479451827811415831, 6.82040439310842801323446435523, 7.95242503029271611384952390383, 8.633791576654839098879316895622, 10.146504377904502603546032369133, 11.00160255187141559069367562044, 12.07978251316481169283168428070, 12.8688718012282192224075374356, 14.15493953764930907119540953545, 15.07802086846011503520132696349, 16.07485182345185855757686102052, 16.69591993747594918955369087129, 18.1805770584210410384149611515, 18.858704892411187039164447133329, 19.76404353126911888963535059653, 20.77442184050251988958964815726, 21.5982365837241922073647016725, 22.81160989003687768722095237502, 23.68454348055760375293784637674, 24.063511886360977504149515471340, 25.6056843492605310625696592620, 26.21322969378816793788829813363

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