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Properties

Label 2-136-136.101-c1-0-4
Degree $2$
Conductor $136$
Sign $0.997 + 0.0676i$
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  + (−1.25 − 0.658i)2-s + 0.909·3-s + (1.13 + 1.64i)4-s + 1.54·5-s + (−1.13 − 0.598i)6-s + 3.57i·7-s + (−0.334 − 2.80i)8-s − 2.17·9-s + (−1.94 − 1.02i)10-s + 5.24·11-s + (1.03 + 1.49i)12-s − 0.927i·13-s + (2.35 − 4.47i)14-s + 1.40·15-s + (−1.42 + 3.73i)16-s + (0.764 − 4.05i)17-s + ⋯
L(s)  = 1  + (−0.885 − 0.465i)2-s + 0.524·3-s + (0.566 + 0.823i)4-s + 0.693·5-s + (−0.464 − 0.244i)6-s + 1.35i·7-s + (−0.118 − 0.992i)8-s − 0.724·9-s + (−0.613 − 0.322i)10-s + 1.58·11-s + (0.297 + 0.432i)12-s − 0.257i·13-s + (0.628 − 1.19i)14-s + 0.363·15-s + (−0.357 + 0.933i)16-s + (0.185 − 0.982i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0676i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.997 + 0.0676i$
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: \((2,\ 136,\ (\ :1/2),\ 0.997 + 0.0676i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.953283 - 0.0322823i\)
\(L(\frac12)\) \(\approx\) \(0.953283 - 0.0322823i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.25 + 0.658i)T \)
17 \( 1 + (-0.764 + 4.05i)T \)
good3 \( 1 - 0.909T + 3T^{2} \)
5 \( 1 - 1.54T + 5T^{2} \)
7 \( 1 - 3.57iT - 7T^{2} \)
11 \( 1 - 5.24T + 11T^{2} \)
13 \( 1 + 0.927iT - 13T^{2} \)
19 \( 1 + 5.22iT - 19T^{2} \)
23 \( 1 - 5.37iT - 23T^{2} \)
29 \( 1 + 3.00T + 29T^{2} \)
31 \( 1 + 0.336iT - 31T^{2} \)
37 \( 1 + 7.49T + 37T^{2} \)
41 \( 1 + 5.03iT - 41T^{2} \)
43 \( 1 - 8.91iT - 43T^{2} \)
47 \( 1 + 4.93T + 47T^{2} \)
53 \( 1 + 11.5iT - 53T^{2} \)
59 \( 1 + 4.95iT - 59T^{2} \)
61 \( 1 - 9.31T + 61T^{2} \)
67 \( 1 - 1.27iT - 67T^{2} \)
71 \( 1 - 4.52iT - 71T^{2} \)
73 \( 1 - 9.78iT - 73T^{2} \)
79 \( 1 + 7.76iT - 79T^{2} \)
83 \( 1 + 6.80iT - 83T^{2} \)
89 \( 1 + 0.0474T + 89T^{2} \)
97 \( 1 - 9.54iT - 97T^{2} \)
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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

−13.10902258686552311297599202139, −11.77192265727381116112123675283, −11.43106763472418389635285769870, −9.566854209214917229811117577182, −9.254453751583082466343868166108, −8.366598549005106542031723387280, −6.86901554355139683143612474609, −5.58689057949383678853706703312, −3.28758893143725778477451430157, −2.05659879194996505871957073349, 1.65351478158556439516562581658, 3.86588292939832778368905714883, 5.90187515253002441205854592573, 6.82197477046407600465906989891, 8.066427872966835590235114590906, 9.005672303152981345540779837061, 9.976291521779206205275816386578, 10.80355476290594321979790703688, 12.05176501200275281429097324224, 13.75680644686633607753149749682

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