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Properties

Label 2-140-35.27-c1-0-3
Degree $2$
Conductor $140$
Sign $-0.972 + 0.231i$
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  + (−1.83 − 1.83i)3-s + (−1.83 + 1.28i)5-s + (−1.57 − 2.12i)7-s + 3.70i·9-s − 4.70·11-s + (−1.83 − 1.83i)13-s + (5.70 + i)15-s + (0.737 − 0.737i)17-s + 4.75·19-s + (−1 + 6.77i)21-s + (3.70 − 3.70i)23-s + (1.70 − 4.70i)25-s + (1.28 − 1.28i)27-s + 0.701i·29-s − 8.79i·31-s + ⋯
L(s)  = 1  + (−1.05 − 1.05i)3-s + (−0.818 + 0.574i)5-s + (−0.596 − 0.802i)7-s + 1.23i·9-s − 1.41·11-s + (−0.507 − 0.507i)13-s + (1.47 + 0.258i)15-s + (0.178 − 0.178i)17-s + 1.09·19-s + (−0.218 + 1.47i)21-s + (0.771 − 0.771i)23-s + (0.340 − 0.940i)25-s + (0.247 − 0.247i)27-s + 0.130i·29-s − 1.58i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.231i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 + 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.972 + 0.231i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ -0.972 + 0.231i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0371536 - 0.316890i\)
\(L(\frac12)\) \(\approx\) \(0.0371536 - 0.316890i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (1.83 - 1.28i)T \)
7 \( 1 + (1.57 + 2.12i)T \)
good3 \( 1 + (1.83 + 1.83i)T + 3iT^{2} \)
11 \( 1 + 4.70T + 11T^{2} \)
13 \( 1 + (1.83 + 1.83i)T + 13iT^{2} \)
17 \( 1 + (-0.737 + 0.737i)T - 17iT^{2} \)
19 \( 1 - 4.75T + 19T^{2} \)
23 \( 1 + (-3.70 + 3.70i)T - 23iT^{2} \)
29 \( 1 - 0.701iT - 29T^{2} \)
31 \( 1 + 8.79iT - 31T^{2} \)
37 \( 1 + (3.70 + 3.70i)T + 37iT^{2} \)
41 \( 1 - 4.75iT - 41T^{2} \)
43 \( 1 + (5 - 5i)T - 43iT^{2} \)
47 \( 1 + (8.05 - 8.05i)T - 47iT^{2} \)
53 \( 1 + (-5 + 5i)T - 53iT^{2} \)
59 \( 1 + 4.75T + 59T^{2} \)
61 \( 1 + 9.50iT - 61T^{2} \)
67 \( 1 + (-5 - 5i)T + 67iT^{2} \)
71 \( 1 + 5.40T + 71T^{2} \)
73 \( 1 + (3.11 + 3.11i)T + 73iT^{2} \)
79 \( 1 + 6.70iT - 79T^{2} \)
83 \( 1 + (7.86 + 7.86i)T + 83iT^{2} \)
89 \( 1 - 13.5T + 89T^{2} \)
97 \( 1 + (-5.49 + 5.49i)T - 97iT^{2} \)
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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

−12.76783769773798413500871034312, −11.66724282724207351181734985231, −10.88212550921766558145134847529, −9.963494317534862998882146411332, −7.81655932941647333696072254958, −7.36457206422299903754301449057, −6.29960642786353366912424965233, −4.96258362900178377981288002298, −3.03438204165323222798640335287, −0.35259732372798149814835101820, 3.36113818588333654118962892379, 4.98231636437775924781913158767, 5.43164776908751106122162030743, 7.15091557246106046705825723467, 8.595759866589693990967144264023, 9.684609561956453952818657536221, 10.58207995387680659151250961407, 11.70971890891278758043054604793, 12.23763992988451244390876639566, 13.39779413177966256940665050571

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