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Properties

Label 2-240-5.4-c3-0-17
Degree 22
Conductor 240240
Sign −0.999+0.0160i-0.999 + 0.0160i
Analytic cond. 14.160414.1604
Root an. cond. 3.763033.76303
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3i·3-s + (−0.178 − 11.1i)5-s − 35.0i·7-s − 9·9-s − 25.6·11-s + 37.6i·13-s + (−33.5 + 0.536i)15-s + 95.7i·17-s + 50.8·19-s − 105.·21-s − 110. i·23-s + (−124. + 4i)25-s + 27i·27-s + 54.5·29-s − 198.·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.0160 − 0.999i)5-s − 1.89i·7-s − 0.333·9-s − 0.702·11-s + 0.803i·13-s + (−0.577 + 0.00923i)15-s + 1.36i·17-s + 0.614·19-s − 1.09·21-s − 1.00i·23-s + (−0.999 + 0.0320i)25-s + 0.192i·27-s + 0.349·29-s − 1.14·31-s + ⋯

Functional equation

Λ(s)=(240s/2 ΓC(s) L(s)=((−0.999+0.0160i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0160i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(240s/2 ΓC(s+3/2) L(s)=((−0.999+0.0160i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0160i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 240240    =    24⋅3⋅52^{4} \cdot 3 \cdot 5
Sign: −0.999+0.0160i-0.999 + 0.0160i
Analytic conductor: 14.160414.1604
Root analytic conductor: 3.763033.76303
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ240(49,⋅)\chi_{240} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 240, ( :3/2), −0.999+0.0160i)(2,\ 240,\ (\ :3/2),\ -0.999 + 0.0160i)

Particular Values

L(2)L(2) ≈\approx 0.00888635−1.11058i0.00888635 - 1.11058i
L(12)L(\frac12) ≈\approx 0.00888635−1.11058i0.00888635 - 1.11058i
L(52)L(\frac{5}{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
3 1+3iT 1 + 3iT
5 1+(0.178+11.1i)T 1 + (0.178 + 11.1i)T
good7 1+35.0iT−343T2 1 + 35.0iT - 343T^{2}
11 1+25.6T+1.33e3T2 1 + 25.6T + 1.33e3T^{2}
13 1−37.6iT−2.19e3T2 1 - 37.6iT - 2.19e3T^{2}
17 1−95.7iT−4.91e3T2 1 - 95.7iT - 4.91e3T^{2}
19 1−50.8T+6.85e3T2 1 - 50.8T + 6.85e3T^{2}
23 1+110.iT−1.21e4T2 1 + 110. iT - 1.21e4T^{2}
29 1−54.5T+2.43e4T2 1 - 54.5T + 2.43e4T^{2}
31 1+198.T+2.97e4T2 1 + 198.T + 2.97e4T^{2}
37 1+266.iT−5.06e4T2 1 + 266. iT - 5.06e4T^{2}
41 1−103.T+6.89e4T2 1 - 103.T + 6.89e4T^{2}
43 1+108iT−7.95e4T2 1 + 108iT - 7.95e4T^{2}
47 1−597.iT−1.03e5T2 1 - 597. iT - 1.03e5T^{2}
53 1+305.iT−1.48e5T2 1 + 305. iT - 1.48e5T^{2}
59 1+223.T+2.05e5T2 1 + 223.T + 2.05e5T^{2}
61 1−485.T+2.26e5T2 1 - 485.T + 2.26e5T^{2}
67 1+876.iT−3.00e5T2 1 + 876. iT - 3.00e5T^{2}
71 1+585.T+3.57e5T2 1 + 585.T + 3.57e5T^{2}
73 1+1.13e3iT−3.89e5T2 1 + 1.13e3iT - 3.89e5T^{2}
79 1−685.T+4.93e5T2 1 - 685.T + 4.93e5T^{2}
83 1+305.iT−5.71e5T2 1 + 305. iT - 5.71e5T^{2}
89 1+887.T+7.04e5T2 1 + 887.T + 7.04e5T^{2}
97 1−556.iT−9.12e5T2 1 - 556. iT - 9.12e5T^{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

−11.11175578288673786317611238596, −10.38234440980093782442343390643, −9.195129025820703077377122719733, −8.037510290373641179944365949318, −7.36309866828901167196422079513, −6.18896796102865174884084972502, −4.74796193040842460574158391353, −3.78806937567154481421991053646, −1.67458396370741818752482300532, −0.44017551898848511826201330403, 2.49961775478193313181638303235, 3.22133505303984219599411898940, 5.20595113947110746187314628982, 5.75302322313873772384326035754, 7.21081906028568915162418290393, 8.336417974054437322523780050378, 9.405092260829070861486099379508, 10.13675687228175080325443623427, 11.35114036273896951875872514689, 11.84121038429802159321164183724

Graph of the ZZ-function along the critical line