↑

Properties

Label 2-370-37.26-c1-0-5
Degree 22
Conductor 370370
Sign 0.805+0.592i0.805 + 0.592i
Analytic cond. 2.954462.95446
Root an. cond. 1.718851.71885
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.144 − 0.250i)3-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s − 0.289·6-s + (−1.26 + 2.18i)7-s + 0.999·8-s + (1.45 + 2.52i)9-s − 0.999·10-s + 3.81·11-s + (0.144 + 0.250i)12-s + (2.86 − 4.96i)13-s + 2.52·14-s + (−0.144 − 0.250i)15-s + (−0.5 − 0.866i)16-s + (1.84 + 3.18i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.0834 − 0.144i)3-s + (−0.249 + 0.433i)4-s + (0.223 − 0.387i)5-s − 0.118·6-s + (−0.477 + 0.826i)7-s + 0.353·8-s + (0.486 + 0.841i)9-s − 0.316·10-s + 1.14·11-s + (0.0417 + 0.0722i)12-s + (0.794 − 1.37i)13-s + 0.674·14-s + (−0.0373 − 0.0646i)15-s + (−0.125 − 0.216i)16-s + (0.446 + 0.773i)17-s + ⋯

Functional equation

Λ(s)=(370s/2 ΓC(s) L(s)=((0.805+0.592i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(370s/2 ΓC(s+1/2) L(s)=((0.805+0.592i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 370370    =    2⋅5⋅372 \cdot 5 \cdot 37
Sign: 0.805+0.592i0.805 + 0.592i
Analytic conductor: 2.954462.95446
Root analytic conductor: 1.718851.71885
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ370(211,⋅)\chi_{370} (211, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 370, ( :1/2), 0.805+0.592i)(2,\ 370,\ (\ :1/2),\ 0.805 + 0.592i)

Particular Values

L(1)L(1) ≈\approx 1.19455−0.392191i1.19455 - 0.392191i
L(12)L(\frac12) ≈\approx 1.19455−0.392191i1.19455 - 0.392191i
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+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
5 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
37 1+(−5.15+3.23i)T 1 + (-5.15 + 3.23i)T
good3 1+(−0.144+0.250i)T+(−1.5−2.59i)T2 1 + (-0.144 + 0.250i)T + (-1.5 - 2.59i)T^{2}
7 1+(1.26−2.18i)T+(−3.5−6.06i)T2 1 + (1.26 - 2.18i)T + (-3.5 - 6.06i)T^{2}
11 1−3.81T+11T2 1 - 3.81T + 11T^{2}
13 1+(−2.86+4.96i)T+(−6.5−11.2i)T2 1 + (-2.86 + 4.96i)T + (-6.5 - 11.2i)T^{2}
17 1+(−1.84−3.18i)T+(−8.5+14.7i)T2 1 + (-1.84 - 3.18i)T + (-8.5 + 14.7i)T^{2}
19 1+(−1.64+2.84i)T+(−9.5−16.4i)T2 1 + (-1.64 + 2.84i)T + (-9.5 - 16.4i)T^{2}
23 1+5.81T+23T2 1 + 5.81T + 23T^{2}
29 1−4.20T+29T2 1 - 4.20T + 29T^{2}
31 1−2.44T+31T2 1 - 2.44T + 31T^{2}
41 1+(2.14−3.71i)T+(−20.5−35.5i)T2 1 + (2.14 - 3.71i)T + (-20.5 - 35.5i)T^{2}
43 1−1.33T+43T2 1 - 1.33T + 43T^{2}
47 1+5.64T+47T2 1 + 5.64T + 47T^{2}
53 1+(4.50+7.81i)T+(−26.5+45.8i)T2 1 + (4.50 + 7.81i)T + (-26.5 + 45.8i)T^{2}
59 1+(−1.38−2.39i)T+(−29.5+51.0i)T2 1 + (-1.38 - 2.39i)T + (-29.5 + 51.0i)T^{2}
61 1+(−1.89+3.28i)T+(−30.5−52.8i)T2 1 + (-1.89 + 3.28i)T + (-30.5 - 52.8i)T^{2}
67 1+(5.33−9.24i)T+(−33.5−58.0i)T2 1 + (5.33 - 9.24i)T + (-33.5 - 58.0i)T^{2}
71 1+(3.23−5.60i)T+(−35.5−61.4i)T2 1 + (3.23 - 5.60i)T + (-35.5 - 61.4i)T^{2}
73 1−3.68T+73T2 1 - 3.68T + 73T^{2}
79 1+(4.81−8.33i)T+(−39.5−68.4i)T2 1 + (4.81 - 8.33i)T + (-39.5 - 68.4i)T^{2}
83 1+(2.91+5.05i)T+(−41.5+71.8i)T2 1 + (2.91 + 5.05i)T + (-41.5 + 71.8i)T^{2}
89 1+(−4.06−7.04i)T+(−44.5+77.0i)T2 1 + (-4.06 - 7.04i)T + (-44.5 + 77.0i)T^{2}
97 1+11.4T+97T2 1 + 11.4T + 97T^{2}
show more
show less
   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

−11.31441376524831375384927710456, −10.26554674281918097371873235463, −9.599923605101825963861338388921, −8.535218001079716934709381165942, −7.949166636999463076145621305883, −6.45511118436561752728484633454, −5.46510240700244332980823877952, −4.07845937619389538444402386742, −2.78284069735148578753788491084, −1.35353940923775450100826079347, 1.30779729684796337297059234854, 3.57125942676901183339414649271, 4.37583017006602314302165414779, 6.23509721058419002339828757510, 6.58851666713846557459954857287, 7.56691398108478970641444149011, 8.879932980236489986910520564877, 9.643736220553858693288606754795, 10.19304443816871801402793756513, 11.49460136119246568161317387677

Graph of the ZZ-function along the critical line