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Properties

Label 2-136-136.101-c1-0-8
Degree $2$
Conductor $136$
Sign $0.410 - 0.911i$
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  + (0.288 + 1.38i)2-s + 1.14·3-s + (−1.83 + 0.798i)4-s + 3.30·5-s + (0.329 + 1.58i)6-s − 1.93i·7-s + (−1.63 − 2.30i)8-s − 1.69·9-s + (0.953 + 4.57i)10-s − 3.71·11-s + (−2.09 + 0.913i)12-s + 5.23i·13-s + (2.68 − 0.558i)14-s + 3.78·15-s + (2.72 − 2.92i)16-s + (−2.09 − 3.55i)17-s + ⋯
L(s)  = 1  + (0.203 + 0.978i)2-s + 0.660·3-s + (−0.916 + 0.399i)4-s + 1.47·5-s + (0.134 + 0.646i)6-s − 0.732i·7-s + (−0.577 − 0.816i)8-s − 0.563·9-s + (0.301 + 1.44i)10-s − 1.11·11-s + (−0.605 + 0.263i)12-s + 1.45i·13-s + (0.717 − 0.149i)14-s + 0.976·15-s + (0.681 − 0.732i)16-s + (−0.507 − 0.861i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.410 - 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.410 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.410 - 0.911i$
Analytic conductor: \(1.08596\)
Root analytic conductor: \(1.04209\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :1/2),\ 0.410 - 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22190 + 0.789936i\)
\(L(\frac12)\) \(\approx\) \(1.22190 + 0.789936i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.288 - 1.38i)T \)
17 \( 1 + (2.09 + 3.55i)T \)
good3 \( 1 - 1.14T + 3T^{2} \)
5 \( 1 - 3.30T + 5T^{2} \)
7 \( 1 + 1.93iT - 7T^{2} \)
11 \( 1 + 3.71T + 11T^{2} \)
13 \( 1 - 5.23iT - 13T^{2} \)
19 \( 1 + 2.14iT - 19T^{2} \)
23 \( 1 - 3.05iT - 23T^{2} \)
29 \( 1 - 4.62T + 29T^{2} \)
31 \( 1 + 10.3iT - 31T^{2} \)
37 \( 1 - 2.79T + 37T^{2} \)
41 \( 1 - 7.33iT - 41T^{2} \)
43 \( 1 - 6.21iT - 43T^{2} \)
47 \( 1 + 1.60T + 47T^{2} \)
53 \( 1 + 0.675iT - 53T^{2} \)
59 \( 1 - 2.51iT - 59T^{2} \)
61 \( 1 + 0.507T + 61T^{2} \)
67 \( 1 + 6.58iT - 67T^{2} \)
71 \( 1 - 9.04iT - 71T^{2} \)
73 \( 1 + 4.87iT - 73T^{2} \)
79 \( 1 - 3.27iT - 79T^{2} \)
83 \( 1 - 12.9iT - 83T^{2} \)
89 \( 1 + 13.8T + 89T^{2} \)
97 \( 1 + 10.2iT - 97T^{2} \)
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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

−13.65216928285701899303320072128, −13.15750995109818334607285388670, −11.34686777520382034679981589753, −9.769821842393279901464807482892, −9.266889904211687093861629861329, −8.033723370431323900272987003837, −6.86813728127920416200338133944, −5.78209800751539362975335998918, −4.53397311311547510943542397513, −2.62018215618372998814099071844, 2.18793937953667380552906676629, 3.04637755109999201335552082980, 5.24299215241339186343834826951, 5.91860277031996266776183028922, 8.299623859467073057122364363140, 8.912784936003778278323301193811, 10.21550473632998824993186544985, 10.61446585970946546480116023539, 12.32582516896150846193214144292, 13.02268514916235942257921572252

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