↑

Properties

Label 2-370-185.64-c1-0-9
Degree 22
Conductor 370370
Sign 0.997+0.0645i0.997 + 0.0645i
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 + (−0.686 + 0.396i)3-s + (−0.499 − 0.866i)4-s + (0.686 + 2.12i)5-s + 0.792i·6-s + (3 − 1.73i)7-s − 0.999·8-s + (−1.18 + 2.05i)9-s + (2.18 + 0.469i)10-s + (0.686 + 0.396i)12-s + (2.68 + 4.65i)13-s − 3.46i·14-s + (−1.31 − 1.18i)15-s + (−0.5 + 0.866i)16-s + (2.18 − 3.78i)17-s + (1.18 + 2.05i)18-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.396 + 0.228i)3-s + (−0.249 − 0.433i)4-s + (0.306 + 0.951i)5-s + 0.323i·6-s + (1.13 − 0.654i)7-s − 0.353·8-s + (−0.395 + 0.684i)9-s + (0.691 + 0.148i)10-s + (0.198 + 0.114i)12-s + (0.745 + 1.29i)13-s − 0.925i·14-s + (−0.339 − 0.306i)15-s + (−0.125 + 0.216i)16-s + (0.530 − 0.918i)17-s + (0.279 + 0.484i)18-s + ⋯

Functional equation

Λ(s)=(370s/2 ΓC(s) L(s)=((0.997+0.0645i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0645i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(370s/2 ΓC(s+1/2) L(s)=((0.997+0.0645i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0645i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 370370    =    2⋅5⋅372 \cdot 5 \cdot 37
Sign: 0.997+0.0645i0.997 + 0.0645i
Analytic conductor: 2.954462.95446
Root analytic conductor: 1.718851.71885
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ370(249,⋅)\chi_{370} (249, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 370, ( :1/2), 0.997+0.0645i)(2,\ 370,\ (\ :1/2),\ 0.997 + 0.0645i)

Particular Values

L(1)L(1) ≈\approx 1.60445−0.0518526i1.60445 - 0.0518526i
L(12)L(\frac12) ≈\approx 1.60445−0.0518526i1.60445 - 0.0518526i
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+(−0.686−2.12i)T 1 + (-0.686 - 2.12i)T
37 1+(0.5−6.06i)T 1 + (0.5 - 6.06i)T
good3 1+(0.686−0.396i)T+(1.5−2.59i)T2 1 + (0.686 - 0.396i)T + (1.5 - 2.59i)T^{2}
7 1+(−3+1.73i)T+(3.5−6.06i)T2 1 + (-3 + 1.73i)T + (3.5 - 6.06i)T^{2}
11 1+11T2 1 + 11T^{2}
13 1+(−2.68−4.65i)T+(−6.5+11.2i)T2 1 + (-2.68 - 4.65i)T + (-6.5 + 11.2i)T^{2}
17 1+(−2.18+3.78i)T+(−8.5−14.7i)T2 1 + (-2.18 + 3.78i)T + (-8.5 - 14.7i)T^{2}
19 1+(9.5−16.4i)T2 1 + (9.5 - 16.4i)T^{2}
23 1−8.74T+23T2 1 - 8.74T + 23T^{2}
29 1+4.40iT−29T2 1 + 4.40iT - 29T^{2}
31 1−1.08iT−31T2 1 - 1.08iT - 31T^{2}
41 1+(2.87+4.97i)T+(−20.5+35.5i)T2 1 + (2.87 + 4.97i)T + (-20.5 + 35.5i)T^{2}
43 1+8.11T+43T2 1 + 8.11T + 43T^{2}
47 1+1.58iT−47T2 1 + 1.58iT - 47T^{2}
53 1+(3.68+2.12i)T+(26.5+45.8i)T2 1 + (3.68 + 2.12i)T + (26.5 + 45.8i)T^{2}
59 1+(4.62+2.67i)T+(29.5+51.0i)T2 1 + (4.62 + 2.67i)T + (29.5 + 51.0i)T^{2}
61 1+(6.55−3.78i)T+(30.5−52.8i)T2 1 + (6.55 - 3.78i)T + (30.5 - 52.8i)T^{2}
67 1+(10.1−5.84i)T+(33.5−58.0i)T2 1 + (10.1 - 5.84i)T + (33.5 - 58.0i)T^{2}
71 1+(−4.37−7.57i)T+(−35.5+61.4i)T2 1 + (-4.37 - 7.57i)T + (-35.5 + 61.4i)T^{2}
73 1+6.92iT−73T2 1 + 6.92iT - 73T^{2}
79 1+(−10.1+5.84i)T+(39.5−68.4i)T2 1 + (-10.1 + 5.84i)T + (39.5 - 68.4i)T^{2}
83 1+(−0.255−0.147i)T+(41.5+71.8i)T2 1 + (-0.255 - 0.147i)T + (41.5 + 71.8i)T^{2}
89 1+(11.1+6.45i)T+(44.5+77.0i)T2 1 + (11.1 + 6.45i)T + (44.5 + 77.0i)T^{2}
97 1+5.11T+97T2 1 + 5.11T + 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

−11.31994623580952240518743566041, −10.78041420874929812429039042613, −9.921966588180522620019051828597, −8.748600368663142231991025125243, −7.51460607607085403308578618025, −6.53419079352469640806647787369, −5.26834928535282737808378623846, −4.46485725654270067965964940139, −3.08205489312316413880615313022, −1.66561955281867821016586887475, 1.27443413037725629816542276522, 3.33365569672539308744100477056, 4.92366319039026485112379086987, 5.50737918603935609413401946910, 6.32191831215331909486158835900, 7.79290517627784754484632195737, 8.532374968966764102008181554437, 9.174752830196553745472844247649, 10.68870680257684654269153945978, 11.57756602438763639377211797366

Graph of the ZZ-function along the critical line