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Properties

Label 2-140-140.59-c1-0-15
Degree 22
Conductor 140140
Sign 0.998+0.0490i0.998 + 0.0490i
Analytic cond. 1.117901.11790
Root an. cond. 1.057311.05731
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(140s/2 ΓC(s) L(s)=((0.998+0.0490i) Λ‾(2−s)\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}
Λ(s)=(140s/2 ΓC(s+1/2) L(s)=((0.998+0.0490i) Λ‾(1−s)\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: 22
Conductor: 140140    =    22⋅5⋅72^{2} \cdot 5 \cdot 7
Sign: 0.998+0.0490i0.998 + 0.0490i
Analytic conductor: 1.117901.11790
Root analytic conductor: 1.057311.05731
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ140(59,⋅)\chi_{140} (59, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 140, ( :1/2), 0.998+0.0490i)(2,\ 140,\ (\ :1/2),\ 0.998 + 0.0490i)

Particular Values

L(1)L(1) ≈\approx 1.88837−0.0463220i1.88837 - 0.0463220i
L(12)L(\frac12) ≈\approx 1.88837−0.0463220i1.88837 - 0.0463220i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(−1.36+0.365i)T 1 + (-1.36 + 0.365i)T
5 1+(2.22−0.175i)T 1 + (2.22 - 0.175i)T
7 1+(0.664−2.56i)T 1 + (0.664 - 2.56i)T
good3 1+(−1.28−0.739i)T+(1.5+2.59i)T2 1 + (-1.28 - 0.739i)T + (1.5 + 2.59i)T^{2}
11 1+(5.32+3.07i)T+(5.5+9.52i)T2 1 + (5.32 + 3.07i)T + (5.5 + 9.52i)T^{2}
13 1−3.33T+13T2 1 - 3.33T + 13T^{2}
17 1+(1.27−2.20i)T+(−8.5−14.7i)T2 1 + (1.27 - 2.20i)T + (-8.5 - 14.7i)T^{2}
19 1+(−0.352−0.611i)T+(−9.5+16.4i)T2 1 + (-0.352 - 0.611i)T + (-9.5 + 16.4i)T^{2}
23 1+(−0.983−1.70i)T+(−11.5+19.9i)T2 1 + (-0.983 - 1.70i)T + (-11.5 + 19.9i)T^{2}
29 1−5.17T+29T2 1 - 5.17T + 29T^{2}
31 1+(3.40−5.89i)T+(−15.5−26.8i)T2 1 + (3.40 - 5.89i)T + (-15.5 - 26.8i)T^{2}
37 1+(5.90−3.40i)T+(18.5−32.0i)T2 1 + (5.90 - 3.40i)T + (18.5 - 32.0i)T^{2}
41 1−2.53iT−41T2 1 - 2.53iT - 41T^{2}
43 1−4.59T+43T2 1 - 4.59T + 43T^{2}
47 1+(3.78−2.18i)T+(23.5−40.7i)T2 1 + (3.78 - 2.18i)T + (23.5 - 40.7i)T^{2}
53 1+(−4.80−2.77i)T+(26.5+45.8i)T2 1 + (-4.80 - 2.77i)T + (26.5 + 45.8i)T^{2}
59 1+(−3.40+5.89i)T+(−29.5−51.0i)T2 1 + (-3.40 + 5.89i)T + (-29.5 - 51.0i)T^{2}
61 1+(−3.07+1.77i)T+(30.5−52.8i)T2 1 + (-3.07 + 1.77i)T + (30.5 - 52.8i)T^{2}
67 1+(−1.45+2.51i)T+(−33.5−58.0i)T2 1 + (-1.45 + 2.51i)T + (-33.5 - 58.0i)T^{2}
71 1+3.37iT−71T2 1 + 3.37iT - 71T^{2}
73 1+(−1.27+2.20i)T+(−36.5−63.2i)T2 1 + (-1.27 + 2.20i)T + (-36.5 - 63.2i)T^{2}
79 1+(−5.38+3.10i)T+(39.5−68.4i)T2 1 + (-5.38 + 3.10i)T + (39.5 - 68.4i)T^{2}
83 1+4.70iT−83T2 1 + 4.70iT - 83T^{2}
89 1+(5.19−3.00i)T+(44.5−77.0i)T2 1 + (5.19 - 3.00i)T + (44.5 - 77.0i)T^{2}
97 1−9.46T+97T2 1 - 9.46T + 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line