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Properties

Label 1-136-136.67-r1-0-0
Degree $1$
Conductor $136$
Sign $1$
Analytic cond. $14.6152$
Root an. cond. $14.6152$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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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

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

Invariants

Degree: \(1\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $1$
Analytic conductor: \(14.6152\)
Root analytic conductor: \(14.6152\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{136} (67, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 136,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.660726787\)
\(L(\frac12)\) \(\approx\) \(1.660726787\)
\(L(1)\) \(\approx\) \(1.077557390\)
\(L(1)\) \(\approx\) \(1.077557390\)

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 \( 1 - T \)
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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

−28.60315340891111986160045958536, −27.26767363501776459296788773785, −26.5485241094263231237482265897, −25.02432397503329717350895094124, −24.31128337736635647994148646865, −23.330456000598261096150389715893, −22.17034586716287784745693852886, −21.375155757299623808195178527196, −20.61005051187982235551615767017, −18.85648044101547788271068693320, −17.813471036829807021715352260162, −17.40034064489672093591180759333, −16.22388866106738841129404348355, −14.96454343882562942177786519991, −13.75795130480094997847836263477, −12.68000131791179378447004095504, −11.54185365257519585364848691407, −10.496086649672269068579954784355, −9.62472681158110492070165954874, −7.93285127786370270414114389692, −6.71934241768142324311761146741, −5.31853792591302274688169657994, −4.8508695925357760646497564170, −2.50522352665419220371952851214, −1.034537902244349465519946670018, 1.034537902244349465519946670018, 2.50522352665419220371952851214, 4.8508695925357760646497564170, 5.31853792591302274688169657994, 6.71934241768142324311761146741, 7.93285127786370270414114389692, 9.62472681158110492070165954874, 10.496086649672269068579954784355, 11.54185365257519585364848691407, 12.68000131791179378447004095504, 13.75795130480094997847836263477, 14.96454343882562942177786519991, 16.22388866106738841129404348355, 17.40034064489672093591180759333, 17.813471036829807021715352260162, 18.85648044101547788271068693320, 20.61005051187982235551615767017, 21.375155757299623808195178527196, 22.17034586716287784745693852886, 23.330456000598261096150389715893, 24.31128337736635647994148646865, 25.02432397503329717350895094124, 26.5485241094263231237482265897, 27.26767363501776459296788773785, 28.60315340891111986160045958536

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