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Properties

Label 2-370-185.103-c1-0-16
Degree 22
Conductor 370370
Sign −0.993−0.116i-0.993 - 0.116i
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.758 − 2.83i)3-s + (−0.499 + 0.866i)4-s + (2.23 + 0.0488i)5-s + (−2.07 + 2.07i)6-s + (−0.644 − 2.40i)7-s + 0.999·8-s + (−4.84 + 2.79i)9-s + (−1.07 − 1.96i)10-s − 2.91i·11-s + (2.83 + 0.758i)12-s + (1.47 − 2.54i)13-s + (−1.76 + 1.76i)14-s + (−1.55 − 6.36i)15-s + (−0.5 − 0.866i)16-s + (−3.89 + 2.25i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.438 − 1.63i)3-s + (−0.249 + 0.433i)4-s + (0.999 + 0.0218i)5-s + (−0.846 + 0.846i)6-s + (−0.243 − 0.909i)7-s + 0.353·8-s + (−1.61 + 0.932i)9-s + (−0.340 − 0.619i)10-s − 0.878i·11-s + (0.817 + 0.219i)12-s + (0.407 − 0.706i)13-s + (−0.470 + 0.470i)14-s + (−0.402 − 1.64i)15-s + (−0.125 − 0.216i)16-s + (−0.945 + 0.545i)17-s + ⋯

Functional equation

Λ(s)=(370s/2 ΓC(s) L(s)=((−0.993−0.116i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(370s/2 ΓC(s+1/2) L(s)=((−0.993−0.116i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 370370    =    2⋅5⋅372 \cdot 5 \cdot 37
Sign: −0.993−0.116i-0.993 - 0.116i
Analytic conductor: 2.954462.95446
Root analytic conductor: 1.718851.71885
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ370(103,⋅)\chi_{370} (103, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 370, ( :1/2), −0.993−0.116i)(2,\ 370,\ (\ :1/2),\ -0.993 - 0.116i)

Particular Values

L(1)L(1) ≈\approx 0.0555501+0.953699i0.0555501 + 0.953699i
L(12)L(\frac12) ≈\approx 0.0555501+0.953699i0.0555501 + 0.953699i
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+(−2.23−0.0488i)T 1 + (-2.23 - 0.0488i)T
37 1+(−2.51−5.53i)T 1 + (-2.51 - 5.53i)T
good3 1+(0.758+2.83i)T+(−2.59+1.5i)T2 1 + (0.758 + 2.83i)T + (-2.59 + 1.5i)T^{2}
7 1+(0.644+2.40i)T+(−6.06+3.5i)T2 1 + (0.644 + 2.40i)T + (-6.06 + 3.5i)T^{2}
11 1+2.91iT−11T2 1 + 2.91iT - 11T^{2}
13 1+(−1.47+2.54i)T+(−6.5−11.2i)T2 1 + (-1.47 + 2.54i)T + (-6.5 - 11.2i)T^{2}
17 1+(3.89−2.25i)T+(8.5−14.7i)T2 1 + (3.89 - 2.25i)T + (8.5 - 14.7i)T^{2}
19 1+(3.54−0.949i)T+(16.4−9.5i)T2 1 + (3.54 - 0.949i)T + (16.4 - 9.5i)T^{2}
23 1−6.99T+23T2 1 - 6.99T + 23T^{2}
29 1+(3.87−3.87i)T−29iT2 1 + (3.87 - 3.87i)T - 29iT^{2}
31 1+(−3.48−3.48i)T+31iT2 1 + (-3.48 - 3.48i)T + 31iT^{2}
41 1+(4.86+2.81i)T+(20.5+35.5i)T2 1 + (4.86 + 2.81i)T + (20.5 + 35.5i)T^{2}
43 1+9.87T+43T2 1 + 9.87T + 43T^{2}
47 1+(−6.90+6.90i)T−47iT2 1 + (-6.90 + 6.90i)T - 47iT^{2}
53 1+(−2.56+9.56i)T+(−45.8−26.5i)T2 1 + (-2.56 + 9.56i)T + (-45.8 - 26.5i)T^{2}
59 1+(1.43−5.34i)T+(−51.0−29.5i)T2 1 + (1.43 - 5.34i)T + (-51.0 - 29.5i)T^{2}
61 1+(−4.95+1.32i)T+(52.8−30.5i)T2 1 + (-4.95 + 1.32i)T + (52.8 - 30.5i)T^{2}
67 1+(3.36−0.901i)T+(58.0−33.5i)T2 1 + (3.36 - 0.901i)T + (58.0 - 33.5i)T^{2}
71 1+(−8.12+14.0i)T+(−35.5−61.4i)T2 1 + (-8.12 + 14.0i)T + (-35.5 - 61.4i)T^{2}
73 1+(−1.91+1.91i)T−73iT2 1 + (-1.91 + 1.91i)T - 73iT^{2}
79 1+(−4.93+1.32i)T+(68.4−39.5i)T2 1 + (-4.93 + 1.32i)T + (68.4 - 39.5i)T^{2}
83 1+(−2.48+9.27i)T+(−71.8−41.5i)T2 1 + (-2.48 + 9.27i)T + (-71.8 - 41.5i)T^{2}
89 1+(−5.84−1.56i)T+(77.0+44.5i)T2 1 + (-5.84 - 1.56i)T + (77.0 + 44.5i)T^{2}
97 1−1.22iT−97T2 1 - 1.22iT - 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

−10.79971245060857352001199610297, −10.44427969000285881431365618957, −8.894216822305488481442791998991, −8.211682409408098328662483666586, −6.92628323852528278625050600362, −6.42874989698329017826208311319, −5.23738451505446410672931455715, −3.28776261361521060631460829226, −1.91328394357270392097406505720, −0.77784879515372244757471272205, 2.44454503270605253382489617913, 4.29478923519073251775025133481, 5.08518341372492437844206768259, 5.98685652350658506262433296667, 6.83495785247823697137743660784, 8.696858211738579043784644269704, 9.294624274566049391374884222535, 9.725701538246112377574087351744, 10.75214304167900995239577198372, 11.46621150796145557723147219556

Graph of the ZZ-function along the critical line