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Properties

Label 2-136-136.11-c1-0-10
Degree $2$
Conductor $136$
Sign $0.993 - 0.110i$
Analytic cond. $1.08596$
Root an. cond. $1.04209$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.30 + 0.541i)2-s + (−0.242 − 1.21i)3-s + (1.41 + 1.41i)4-s + (0.342 − 1.72i)6-s + (1.08 + 2.61i)8-s + (1.34 − 0.559i)9-s + (−3.86 − 0.769i)11-s + (1.37 − 2.06i)12-s + 4i·16-s + (−3.85 + 1.46i)17-s + 2.06·18-s + (−4.23 − 1.75i)19-s + (−4.63 − 3.09i)22-s + (2.91 − 1.94i)24-s + (1.91 + 4.61i)25-s + ⋯
L(s)  = 1  + (0.923 + 0.382i)2-s + (−0.139 − 0.702i)3-s + (0.707 + 0.707i)4-s + (0.139 − 0.702i)6-s + (0.382 + 0.923i)8-s + (0.449 − 0.186i)9-s + (−1.16 − 0.232i)11-s + (0.397 − 0.595i)12-s + i·16-s + (−0.934 + 0.355i)17-s + 0.486·18-s + (−0.971 − 0.402i)19-s + (−0.988 − 0.660i)22-s + (0.595 − 0.397i)24-s + (0.382 + 0.923i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.993 - 0.110i$
Analytic conductor: \(1.08596\)
Root analytic conductor: \(1.04209\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :1/2),\ 0.993 - 0.110i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67036 + 0.0921960i\)
\(L(\frac12)\) \(\approx\) \(1.67036 + 0.0921960i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.30 - 0.541i)T \)
17 \( 1 + (3.85 - 1.46i)T \)
good3 \( 1 + (0.242 + 1.21i)T + (-2.77 + 1.14i)T^{2} \)
5 \( 1 + (-1.91 - 4.61i)T^{2} \)
7 \( 1 + (-2.67 + 6.46i)T^{2} \)
11 \( 1 + (3.86 + 0.769i)T + (10.1 + 4.20i)T^{2} \)
13 \( 1 + 13iT^{2} \)
19 \( 1 + (4.23 + 1.75i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (21.2 + 8.80i)T^{2} \)
29 \( 1 + (11.0 + 26.7i)T^{2} \)
31 \( 1 + (-28.6 + 11.8i)T^{2} \)
37 \( 1 + (34.1 - 14.1i)T^{2} \)
41 \( 1 + (-3.05 + 2.04i)T + (15.6 - 37.8i)T^{2} \)
43 \( 1 + (-11.5 + 4.77i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-0.0502 - 0.121i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (23.3 - 56.3i)T^{2} \)
67 \( 1 - 13.1iT - 67T^{2} \)
71 \( 1 + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (-14.1 - 9.46i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (-72.9 - 30.2i)T^{2} \)
83 \( 1 + (1.63 - 3.94i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (13.2 + 13.2i)T + 89iT^{2} \)
97 \( 1 + (-9.85 + 14.7i)T + (-37.1 - 89.6i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.90744527313113014590482191346, −12.84308446399532245477271941787, −11.42725360543659540117312243250, −10.51506003359611382523429911523, −8.703181992409087782905109307635, −7.53613095855622993517830871498, −6.68721586619529417129240848443, −5.52485935603065002299893734233, −4.15046397426586435517510587630, −2.36875152739315199465400906459, 2.45304788197585007639762459122, 4.17816397323458371809694233499, 4.99443519482791595583765232423, 6.34216174085863634458052396472, 7.70819103284721025838214162880, 9.407525184629720139059768079045, 10.52931887603928652789172523854, 10.93976608814523361367385320086, 12.40942645435237215434852440646, 13.07618570206919149933350815347

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