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Properties

Label 2-140-7.4-c1-0-0
Degree $2$
Conductor $140$
Sign $0.701 - 0.712i$
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  + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)5-s + (2.5 − 0.866i)7-s + (1 + 1.73i)9-s + (−3 + 5.19i)11-s + 2·13-s − 0.999·15-s + (3 − 5.19i)17-s + (−4 − 6.92i)19-s + (−0.500 + 2.59i)21-s + (−1.5 − 2.59i)23-s + (−0.499 + 0.866i)25-s − 5·27-s + 3·29-s + (−1 + 1.73i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.223 + 0.387i)5-s + (0.944 − 0.327i)7-s + (0.333 + 0.577i)9-s + (−0.904 + 1.56i)11-s + 0.554·13-s − 0.258·15-s + (0.727 − 1.26i)17-s + (−0.917 − 1.58i)19-s + (−0.109 + 0.566i)21-s + (−0.312 − 0.541i)23-s + (−0.0999 + 0.173i)25-s − 0.962·27-s + 0.557·29-s + (−0.179 + 0.311i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.701 - 0.712i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ 0.701 - 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02035 + 0.427612i\)
\(L(\frac12)\) \(\approx\) \(1.02035 + 0.427612i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-2.5 + 0.866i)T \)
good3 \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4 + 6.92i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6 - 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-5 + 8.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3T + 83T^{2} \)
89 \( 1 + (-1.5 - 2.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−13.36706048412841757629063522976, −12.22731696367484953935176051160, −10.90730112731691485270055581781, −10.49487441071752086039410885698, −9.338167679084910960978160382074, −7.82713327274107256128074261923, −6.99365084362842027402329299925, −5.17040697582510466059222722227, −4.48615001777838913573719204557, −2.30107921941669512543073014083, 1.50858264136031691399360890224, 3.72731121181989691548103376156, 5.53064951770157521542506418499, 6.20864127638802650703356683793, 8.064615327217175668888713984766, 8.444227188983535485232190633415, 10.10354980205228988436042381203, 11.08619625972759325409613725196, 12.12519234019566390850642585439, 12.92537522565612462755954713376

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