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Properties

Label 2-312-104.77-c3-0-28
Degree 22
Conductor 312312
Sign −0.460−0.887i-0.460 - 0.887i
Analytic cond. 18.408518.4085
Root an. cond. 4.290524.29052
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  + (−1.04 + 2.62i)2-s + 3i·3-s + (−5.81 − 5.49i)4-s − 7.49·5-s + (−7.88 − 3.13i)6-s − 9.78i·7-s + (20.5 − 9.53i)8-s − 9·9-s + (7.83 − 19.6i)10-s + 72.0·11-s + (16.4 − 17.4i)12-s + (−46.8 + 2.05i)13-s + (25.7 + 10.2i)14-s − 22.4i·15-s + (3.59 + 63.8i)16-s + 14.0·17-s + ⋯
L(s)  = 1  + (−0.369 + 0.929i)2-s + 0.577i·3-s + (−0.726 − 0.686i)4-s − 0.670·5-s + (−0.536 − 0.213i)6-s − 0.528i·7-s + (0.906 − 0.421i)8-s − 0.333·9-s + (0.247 − 0.622i)10-s + 1.97·11-s + (0.396 − 0.419i)12-s + (−0.999 + 0.0439i)13-s + (0.490 + 0.195i)14-s − 0.386i·15-s + (0.0561 + 0.998i)16-s + 0.200·17-s + ⋯

Functional equation

Λ(s)=(312s/2 ΓC(s) L(s)=((−0.460−0.887i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.460 - 0.887i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(312s/2 ΓC(s+3/2) L(s)=((−0.460−0.887i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 312312    =    23⋅3⋅132^{3} \cdot 3 \cdot 13
Sign: −0.460−0.887i-0.460 - 0.887i
Analytic conductor: 18.408518.4085
Root analytic conductor: 4.290524.29052
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ312(181,⋅)\chi_{312} (181, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 312, ( :3/2), −0.460−0.887i)(2,\ 312,\ (\ :3/2),\ -0.460 - 0.887i)

Particular Values

L(2)L(2) ≈\approx 1.1845051231.184505123
L(12)L(\frac12) ≈\approx 1.1845051231.184505123
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+(1.04−2.62i)T 1 + (1.04 - 2.62i)T
3 1−3iT 1 - 3iT
13 1+(46.8−2.05i)T 1 + (46.8 - 2.05i)T
good5 1+7.49T+125T2 1 + 7.49T + 125T^{2}
7 1+9.78iT−343T2 1 + 9.78iT - 343T^{2}
11 1−72.0T+1.33e3T2 1 - 72.0T + 1.33e3T^{2}
17 1−14.0T+4.91e3T2 1 - 14.0T + 4.91e3T^{2}
19 1−126.T+6.85e3T2 1 - 126.T + 6.85e3T^{2}
23 1+74.7T+1.21e4T2 1 + 74.7T + 1.21e4T^{2}
29 1−177.iT−2.43e4T2 1 - 177. iT - 2.43e4T^{2}
31 1−204.iT−2.97e4T2 1 - 204. iT - 2.97e4T^{2}
37 1−262.T+5.06e4T2 1 - 262.T + 5.06e4T^{2}
41 1+376.iT−6.89e4T2 1 + 376. iT - 6.89e4T^{2}
43 1−179.iT−7.95e4T2 1 - 179. iT - 7.95e4T^{2}
47 1−4.21iT−1.03e5T2 1 - 4.21iT - 1.03e5T^{2}
53 1−102.iT−1.48e5T2 1 - 102. iT - 1.48e5T^{2}
59 1−231.T+2.05e5T2 1 - 231.T + 2.05e5T^{2}
61 1−876.iT−2.26e5T2 1 - 876. iT - 2.26e5T^{2}
67 1+597.T+3.00e5T2 1 + 597.T + 3.00e5T^{2}
71 1−576.iT−3.57e5T2 1 - 576. iT - 3.57e5T^{2}
73 1−657.iT−3.89e5T2 1 - 657. iT - 3.89e5T^{2}
79 1+137.T+4.93e5T2 1 + 137.T + 4.93e5T^{2}
83 1−1.08e3T+5.71e5T2 1 - 1.08e3T + 5.71e5T^{2}
89 1+269.iT−7.04e5T2 1 + 269. iT - 7.04e5T^{2}
97 1−1.73e3iT−9.12e5T2 1 - 1.73e3iT - 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

−11.53947567895673986035317535469, −10.30663427965901806444364974154, −9.513674458265010595060923729163, −8.791317198646860638801246243497, −7.54989052839127265544126281062, −6.94116694288333441326695923553, −5.65585945130952517219206984440, −4.45903637342246737480387136350, −3.65635492441444926701992801445, −1.06249392947142418816384505738, 0.64972694753019423033842158365, 2.03487049054812027590155232920, 3.41703682906410060581237309132, 4.47758480108955316600700040135, 6.03996142586831235402914977926, 7.42073823198605277973640575224, 8.072907015690593536248147135230, 9.340838262681964120025918721630, 9.741027373783726730419614075139, 11.45683992653048152170014871932

Graph of the ZZ-function along the critical line