↑

Properties

Label 1-136-136.101-r0-0-0
Degree $1$
Conductor $136$
Sign $1$
Analytic cond. $0.631581$
Root an. cond. $0.631581$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s − 7-s + 9-s + 11-s − 13-s + 15-s − 19-s − 21-s − 23-s + 25-s + 27-s + 29-s − 31-s + 33-s − 35-s + 37-s − 39-s − 41-s − 43-s + 45-s + 47-s + 49-s − 53-s + 55-s − 57-s − 59-s + ⋯
L(s)  = 1  + 3-s + 5-s − 7-s + 9-s + 11-s − 13-s + 15-s − 19-s − 21-s − 23-s + 25-s + 27-s + 29-s − 31-s + 33-s − 35-s + 37-s − 39-s − 41-s − 43-s + 45-s + 47-s + 49-s − 53-s + 55-s − 57-s − 59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $1$
Analytic conductor: \(0.631581\)
Root analytic conductor: \(0.631581\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{136} (101, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 136,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.565959802\)
\(L(\frac12)\) \(\approx\) \(1.565959802\)
\(L(1)\) \(\approx\) \(1.457151825\)
\(L(1)\) \(\approx\) \(1.457151825\)

Euler product

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

−28.68404081445512164791737876591, −27.33023854636295342534610583217, −26.36621626418345757090190646875, −25.35546175131352008928231632289, −25.04316994067640521704661576259, −23.72564304014311803863331319866, −22.05829635263788718136904356299, −21.75914128983445403914986864357, −20.29170332121643428566157855807, −19.59641076495500690070427874899, −18.61377781551928162778894894257, −17.309927217566412297723478339090, −16.32724646566848456202196144949, −14.947593972311764276615303434739, −14.14757675383255549272809575764, −13.162365431062265257246167860300, −12.25011670105187166229263472295, −10.25659748416673504034794408536, −9.580925209380395515540201289, −8.66583580329093748214223264764, −7.08348898872446162809741343739, −6.132983694125622366769591600344, −4.364436884900257673752153440121, −2.99116446940677859873639032969, −1.84328295784507981166774605058, 1.84328295784507981166774605058, 2.99116446940677859873639032969, 4.364436884900257673752153440121, 6.132983694125622366769591600344, 7.08348898872446162809741343739, 8.66583580329093748214223264764, 9.580925209380395515540201289, 10.25659748416673504034794408536, 12.25011670105187166229263472295, 13.162365431062265257246167860300, 14.14757675383255549272809575764, 14.947593972311764276615303434739, 16.32724646566848456202196144949, 17.309927217566412297723478339090, 18.61377781551928162778894894257, 19.59641076495500690070427874899, 20.29170332121643428566157855807, 21.75914128983445403914986864357, 22.05829635263788718136904356299, 23.72564304014311803863331319866, 25.04316994067640521704661576259, 25.35546175131352008928231632289, 26.36621626418345757090190646875, 27.33023854636295342534610583217, 28.68404081445512164791737876591

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