Properties

Label 2-140-28.27-c1-0-14
Degree $2$
Conductor $140$
Sign $0.869 + 0.494i$
Analytic cond. $1.11790$
Root an. cond. $1.05731$
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  + (1.28 − 0.599i)2-s + 0.936·3-s + (1.28 − 1.53i)4-s + i·5-s + (1.19 − 0.561i)6-s + (−2.60 + 0.468i)7-s + (0.719 − 2.73i)8-s − 2.12·9-s + (0.599 + 1.28i)10-s + 2.39i·11-s + (1.19 − 1.43i)12-s + 2i·13-s + (−3.05 + 2.16i)14-s + 0.936i·15-s + (−0.719 − 3.93i)16-s − 7.12i·17-s + ⋯
L(s)  = 1  + (0.905 − 0.424i)2-s + 0.540·3-s + (0.640 − 0.768i)4-s + 0.447i·5-s + (0.489 − 0.229i)6-s + (−0.984 + 0.176i)7-s + (0.254 − 0.967i)8-s − 0.707·9-s + (0.189 + 0.405i)10-s + 0.723i·11-s + (0.346 − 0.415i)12-s + 0.554i·13-s + (−0.816 + 0.577i)14-s + 0.241i·15-s + (−0.179 − 0.983i)16-s − 1.72i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 + 0.494i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.869 + 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.869 + 0.494i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ 0.869 + 0.494i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77092 - 0.468353i\)
\(L(\frac12)\) \(\approx\) \(1.77092 - 0.468353i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.28 + 0.599i)T \)
5 \( 1 - iT \)
7 \( 1 + (2.60 - 0.468i)T \)
good3 \( 1 - 0.936T + 3T^{2} \)
11 \( 1 - 2.39iT - 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 7.12iT - 17T^{2} \)
19 \( 1 - 2.39T + 19T^{2} \)
23 \( 1 - 5.73iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 6.67T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - 7.12iT - 41T^{2} \)
43 \( 1 + 7.60iT - 43T^{2} \)
47 \( 1 + 10.0T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 10.9T + 59T^{2} \)
61 \( 1 - 2iT - 61T^{2} \)
67 \( 1 + 14.2iT - 67T^{2} \)
71 \( 1 - 6.14iT - 71T^{2} \)
73 \( 1 + 9.36iT - 73T^{2} \)
79 \( 1 + 4.27iT - 79T^{2} \)
83 \( 1 - 0.936T + 83T^{2} \)
89 \( 1 - 12iT - 89T^{2} \)
97 \( 1 - 7.12iT - 97T^{2} \)
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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

−13.35715193452817719441512953025, −11.98174159820559750101520720158, −11.42765847871728283033085848812, −9.894828267619128218742900518243, −9.336047205428287307280005866484, −7.45113572907467101968944943252, −6.45518617667683292029523023616, −5.13302341777082330521003232403, −3.49272063160832446146007279785, −2.52697532430818881695725016538, 2.85398824592260804443183247156, 3.93193478839480476454931162721, 5.61147730785555401941343622108, 6.46831283430676715366060310530, 8.036025060601533908303159213417, 8.692895077291286487287511184938, 10.23354193736687729885812485097, 11.46868064180587207923986383908, 12.67587225728130501601433445471, 13.21199995395032955297357356979

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