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Properties

Label 2-140-35.27-c1-0-1
Degree $2$
Conductor $140$
Sign $0.864 - 0.502i$
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 + (−2.12 − 1.57i)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 + (−0.802 − 0.596i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 - 0.502i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.864 - 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.864 - 0.502i$
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: \((2,\ 140,\ (\ :1/2),\ 0.864 - 0.502i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41239 + 0.380630i\)
\(L(\frac12)\) \(\approx\) \(1.41239 + 0.380630i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.12 + 1.57i)T \)
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

−13.36524350662671090945724301450, −12.66237266482079676117666363370, −10.51617259718594155396952195360, −10.28380098007988196146338209255, −9.015491147038210403363873205195, −8.478725231461450239474693836967, −6.77436964694826724823060222175, −5.19811607378322860275989158549, −3.98015117411311510235667276860, −2.57525560097380973823581226295, 2.25113349903304211519715936473, 3.09141158211846384034313085558, 5.62853703861753949669252704998, 6.68080384768345959117319551540, 7.77430473016055535780929785601, 8.788629150391081198743487598626, 9.838974286769616649024000300239, 10.94502901845351706545417865947, 12.58606096651175865594563937661, 13.25844168196566725543234625313

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