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Properties

Label 2-140-140.59-c1-0-15
Degree $2$
Conductor $140$
Sign $0.998 + 0.0490i$
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  + (1.36 − 0.365i)2-s + (1.28 + 0.739i)3-s + (1.73 − 0.998i)4-s + (−2.22 + 0.175i)5-s + (2.02 + 0.542i)6-s + (−0.664 + 2.56i)7-s + (2.00 − 1.99i)8-s + (−0.406 − 0.703i)9-s + (−2.98 + 1.05i)10-s + (−5.32 − 3.07i)11-s + (2.95 + 0.00185i)12-s + 3.33·13-s + (0.0291 + 3.74i)14-s + (−2.98 − 1.42i)15-s + (2.00 − 3.46i)16-s + (−1.27 + 2.20i)17-s + ⋯
L(s)  = 1  + (0.966 − 0.258i)2-s + (0.739 + 0.426i)3-s + (0.866 − 0.499i)4-s + (−0.996 + 0.0784i)5-s + (0.824 + 0.221i)6-s + (−0.250 + 0.967i)7-s + (0.707 − 0.706i)8-s + (−0.135 − 0.234i)9-s + (−0.942 + 0.333i)10-s + (−1.60 − 0.927i)11-s + (0.853 + 0.000534i)12-s + 0.924·13-s + (0.00779 + 0.999i)14-s + (−0.770 − 0.367i)15-s + (0.501 − 0.865i)16-s + (−0.309 + 0.536i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0490i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.998 + 0.0490i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ 0.998 + 0.0490i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88837 - 0.0463220i\)
\(L(\frac12)\) \(\approx\) \(1.88837 - 0.0463220i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 + 0.365i)T \)
5 \( 1 + (2.22 - 0.175i)T \)
7 \( 1 + (0.664 - 2.56i)T \)
good3 \( 1 + (-1.28 - 0.739i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (5.32 + 3.07i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.33T + 13T^{2} \)
17 \( 1 + (1.27 - 2.20i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.352 - 0.611i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.983 - 1.70i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.17T + 29T^{2} \)
31 \( 1 + (3.40 - 5.89i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.90 - 3.40i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.53iT - 41T^{2} \)
43 \( 1 - 4.59T + 43T^{2} \)
47 \( 1 + (3.78 - 2.18i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.80 - 2.77i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.40 + 5.89i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.07 + 1.77i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.45 + 2.51i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.37iT - 71T^{2} \)
73 \( 1 + (-1.27 + 2.20i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.38 + 3.10i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.70iT - 83T^{2} \)
89 \( 1 + (5.19 - 3.00i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.46T + 97T^{2} \)
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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

−13.15451963314600893542644497381, −12.25645835173303484484064291993, −11.24367273461287545476153702418, −10.39044078089492749976062530967, −8.802997885883382868649605759612, −8.030143451073897763047178071768, −6.36514841783929177849536353372, −5.16307643496821428580936307384, −3.59594005310440606460960922845, −2.86778650562995855995279332441, 2.61595154484844340083115413150, 3.92691805093723008093618327016, 5.12517419907809434159620023122, 7.00459471247330379317166017274, 7.60946595219103169455584482321, 8.469871741257807418333365888363, 10.45301280461635587068918324023, 11.25137204570767211372220717932, 12.60581043135230878570526730359, 13.23591571571264733260140220168

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