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Properties

Label 2-312-104.77-c3-0-20
Degree 22
Conductor 312312
Sign 0.981+0.191i0.981 + 0.191i
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  + (0.800 − 2.71i)2-s − 3i·3-s + (−6.71 − 4.34i)4-s − 12.1·5-s + (−8.13 − 2.40i)6-s + 21.1i·7-s + (−17.1 + 14.7i)8-s − 9·9-s + (−9.70 + 32.8i)10-s − 16.0·11-s + (−13.0 + 20.1i)12-s + (36.8 − 29.0i)13-s + (57.3 + 16.9i)14-s + 36.3i·15-s + (26.2 + 58.3i)16-s + 45.8·17-s + ⋯
L(s)  = 1  + (0.283 − 0.959i)2-s − 0.577i·3-s + (−0.839 − 0.543i)4-s − 1.08·5-s + (−0.553 − 0.163i)6-s + 1.14i·7-s + (−0.758 + 0.651i)8-s − 0.333·9-s + (−0.307 + 1.04i)10-s − 0.438·11-s + (−0.313 + 0.484i)12-s + (0.785 − 0.619i)13-s + (1.09 + 0.323i)14-s + 0.626i·15-s + (0.410 + 0.911i)16-s + 0.653·17-s + ⋯

Functional equation

Λ(s)=(312s/2 ΓC(s) L(s)=((0.981+0.191i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(312s/2 ΓC(s+3/2) L(s)=((0.981+0.191i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 312312    =    23⋅3⋅132^{3} \cdot 3 \cdot 13
Sign: 0.981+0.191i0.981 + 0.191i
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), 0.981+0.191i)(2,\ 312,\ (\ :3/2),\ 0.981 + 0.191i)

Particular Values

L(2)L(2) ≈\approx 1.1082130211.108213021
L(12)L(\frac12) ≈\approx 1.1082130211.108213021
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+(−0.800+2.71i)T 1 + (-0.800 + 2.71i)T
3 1+3iT 1 + 3iT
13 1+(−36.8+29.0i)T 1 + (-36.8 + 29.0i)T
good5 1+12.1T+125T2 1 + 12.1T + 125T^{2}
7 1−21.1iT−343T2 1 - 21.1iT - 343T^{2}
11 1+16.0T+1.33e3T2 1 + 16.0T + 1.33e3T^{2}
17 1−45.8T+4.91e3T2 1 - 45.8T + 4.91e3T^{2}
19 1−117.T+6.85e3T2 1 - 117.T + 6.85e3T^{2}
23 1+97.3T+1.21e4T2 1 + 97.3T + 1.21e4T^{2}
29 1−90.0iT−2.43e4T2 1 - 90.0iT - 2.43e4T^{2}
31 1−151.iT−2.97e4T2 1 - 151. iT - 2.97e4T^{2}
37 1−66.3T+5.06e4T2 1 - 66.3T + 5.06e4T^{2}
41 1−112.iT−6.89e4T2 1 - 112. iT - 6.89e4T^{2}
43 1+234.iT−7.95e4T2 1 + 234. iT - 7.95e4T^{2}
47 1−580.iT−1.03e5T2 1 - 580. iT - 1.03e5T^{2}
53 1−241.iT−1.48e5T2 1 - 241. iT - 1.48e5T^{2}
59 1+313.T+2.05e5T2 1 + 313.T + 2.05e5T^{2}
61 1−334.iT−2.26e5T2 1 - 334. iT - 2.26e5T^{2}
67 1−315.T+3.00e5T2 1 - 315.T + 3.00e5T^{2}
71 1−503.iT−3.57e5T2 1 - 503. iT - 3.57e5T^{2}
73 1−757.iT−3.89e5T2 1 - 757. iT - 3.89e5T^{2}
79 1−1.13e3T+4.93e5T2 1 - 1.13e3T + 4.93e5T^{2}
83 1−822.T+5.71e5T2 1 - 822.T + 5.71e5T^{2}
89 1+803.iT−7.04e5T2 1 + 803. iT - 7.04e5T^{2}
97 1−750.iT−9.12e5T2 1 - 750. 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

−11.50814569656800093319868071667, −10.56096707133710763364110041882, −9.359461778270892024458268776846, −8.371983321286301091237631080404, −7.67109629916802693813080753099, −5.97846728247275241569707987319, −5.16315605146332413448515485369, −3.63486254602505085011552770787, −2.73514982208916658908314234438, −1.12670991228808916640224381920, 0.45549523602458334076390992220, 3.54828017135141183636477276196, 4.03281499983128230546853283800, 5.16614532847081868814717754864, 6.42472824895634000112657444436, 7.66637877493551350487542485494, 7.958153594103724398073692045993, 9.308019887405266388811556703074, 10.21080278975245015950399540488, 11.38726237930981107812993573050

Graph of the ZZ-function along the critical line