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Properties

Label 2-312-104.77-c3-0-25
Degree 22
Conductor 312312
Sign −0.996−0.0860i-0.996 - 0.0860i
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  + (2.48 + 1.34i)2-s + 3i·3-s + (4.36 + 6.70i)4-s − 8.37·5-s + (−4.04 + 7.46i)6-s + 16.5i·7-s + (1.83 + 22.5i)8-s − 9·9-s + (−20.8 − 11.2i)10-s + 23.2·11-s + (−20.1 + 13.1i)12-s + (46.2 + 7.80i)13-s + (−22.3 + 41.2i)14-s − 25.1i·15-s + (−25.8 + 58.5i)16-s − 14.7·17-s + ⋯
L(s)  = 1  + (0.879 + 0.476i)2-s + 0.577i·3-s + (0.546 + 0.837i)4-s − 0.748·5-s + (−0.275 + 0.507i)6-s + 0.896i·7-s + (0.0811 + 0.996i)8-s − 0.333·9-s + (−0.658 − 0.356i)10-s + 0.636·11-s + (−0.483 + 0.315i)12-s + (0.986 + 0.166i)13-s + (−0.426 + 0.787i)14-s − 0.432i·15-s + (−0.403 + 0.914i)16-s − 0.210·17-s + ⋯

Functional equation

Λ(s)=(312s/2 ΓC(s) L(s)=((−0.996−0.0860i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0860i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(312s/2 ΓC(s+3/2) L(s)=((−0.996−0.0860i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0860i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 312312    =    23⋅3⋅132^{3} \cdot 3 \cdot 13
Sign: −0.996−0.0860i-0.996 - 0.0860i
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), −0.996−0.0860i)(2,\ 312,\ (\ :3/2),\ -0.996 - 0.0860i)

Particular Values

L(2)L(2) ≈\approx 2.1634082462.163408246
L(12)L(\frac12) ≈\approx 2.1634082462.163408246
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+(−2.48−1.34i)T 1 + (-2.48 - 1.34i)T
3 1−3iT 1 - 3iT
13 1+(−46.2−7.80i)T 1 + (-46.2 - 7.80i)T
good5 1+8.37T+125T2 1 + 8.37T + 125T^{2}
7 1−16.5iT−343T2 1 - 16.5iT - 343T^{2}
11 1−23.2T+1.33e3T2 1 - 23.2T + 1.33e3T^{2}
17 1+14.7T+4.91e3T2 1 + 14.7T + 4.91e3T^{2}
19 1+124.T+6.85e3T2 1 + 124.T + 6.85e3T^{2}
23 1+173.T+1.21e4T2 1 + 173.T + 1.21e4T^{2}
29 1+126.iT−2.43e4T2 1 + 126. iT - 2.43e4T^{2}
31 1+27.9iT−2.97e4T2 1 + 27.9iT - 2.97e4T^{2}
37 1−362.T+5.06e4T2 1 - 362.T + 5.06e4T^{2}
41 1−309.iT−6.89e4T2 1 - 309. iT - 6.89e4T^{2}
43 1−556.iT−7.95e4T2 1 - 556. iT - 7.95e4T^{2}
47 1+163.iT−1.03e5T2 1 + 163. iT - 1.03e5T^{2}
53 1+334.iT−1.48e5T2 1 + 334. iT - 1.48e5T^{2}
59 1−809.T+2.05e5T2 1 - 809.T + 2.05e5T^{2}
61 1+6.20iT−2.26e5T2 1 + 6.20iT - 2.26e5T^{2}
67 1−252.T+3.00e5T2 1 - 252.T + 3.00e5T^{2}
71 1−822.iT−3.57e5T2 1 - 822. iT - 3.57e5T^{2}
73 1−449.iT−3.89e5T2 1 - 449. iT - 3.89e5T^{2}
79 1+547.T+4.93e5T2 1 + 547.T + 4.93e5T^{2}
83 1−959.T+5.71e5T2 1 - 959.T + 5.71e5T^{2}
89 1−656.iT−7.04e5T2 1 - 656. iT - 7.04e5T^{2}
97 1−417.iT−9.12e5T2 1 - 417. 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

−11.52879874462655633069649771878, −11.34681939070531383705730210832, −9.788313616366043376070553090631, −8.511456514363158702171059401776, −8.054899234184950866309436433126, −6.44569484078351549174391106035, −5.87303108264135567945577077580, −4.37010615724869702824724532413, −3.84714088051075071488881255599, −2.34929645918073698974987757608, 0.57432837841630311973721317642, 1.98616956370822765938944813801, 3.72361737186202038213536876384, 4.21248545600705779207786940540, 5.88109479339875727902185173974, 6.71480831637061740037282764632, 7.69816710341010240444017087584, 8.828554111890545078075309473181, 10.31249880282825864895902685794, 10.97755155254204262398737829132

Graph of the ZZ-function along the critical line