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Properties

Label 2-312-104.77-c3-0-35
Degree 22
Conductor 312312
Sign −0.639−0.768i-0.639 - 0.768i
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  + (2.56 + 1.18i)2-s + 3i·3-s + (5.20 + 6.07i)4-s + 8.43·5-s + (−3.54 + 7.70i)6-s + 8.36i·7-s + (6.20 + 21.7i)8-s − 9·9-s + (21.6 + 9.97i)10-s − 57.7·11-s + (−18.2 + 15.6i)12-s + (18.9 + 42.8i)13-s + (−9.88 + 21.5i)14-s + 25.3i·15-s + (−9.78 + 63.2i)16-s − 72.1·17-s + ⋯
L(s)  = 1  + (0.908 + 0.417i)2-s + 0.577i·3-s + (0.650 + 0.759i)4-s + 0.754·5-s + (−0.241 + 0.524i)6-s + 0.451i·7-s + (0.274 + 0.961i)8-s − 0.333·9-s + (0.685 + 0.315i)10-s − 1.58·11-s + (−0.438 + 0.375i)12-s + (0.404 + 0.914i)13-s + (−0.188 + 0.410i)14-s + 0.435i·15-s + (−0.152 + 0.988i)16-s − 1.03·17-s + ⋯

Functional equation

Λ(s)=(312s/2 ΓC(s) L(s)=((−0.639−0.768i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.639 - 0.768i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(312s/2 ΓC(s+3/2) L(s)=((−0.639−0.768i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.639 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 312312    =    23⋅3⋅132^{3} \cdot 3 \cdot 13
Sign: −0.639−0.768i-0.639 - 0.768i
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), −0.639−0.768i)(2,\ 312,\ (\ :3/2),\ -0.639 - 0.768i)

Particular Values

L(2)L(2) ≈\approx 3.2481112193.248111219
L(12)L(\frac12) ≈\approx 3.2481112193.248111219
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+(−2.56−1.18i)T 1 + (-2.56 - 1.18i)T
3 1−3iT 1 - 3iT
13 1+(−18.9−42.8i)T 1 + (-18.9 - 42.8i)T
good5 1−8.43T+125T2 1 - 8.43T + 125T^{2}
7 1−8.36iT−343T2 1 - 8.36iT - 343T^{2}
11 1+57.7T+1.33e3T2 1 + 57.7T + 1.33e3T^{2}
17 1+72.1T+4.91e3T2 1 + 72.1T + 4.91e3T^{2}
19 1−144.T+6.85e3T2 1 - 144.T + 6.85e3T^{2}
23 1−168.T+1.21e4T2 1 - 168.T + 1.21e4T^{2}
29 1+96.0iT−2.43e4T2 1 + 96.0iT - 2.43e4T^{2}
31 1−59.3iT−2.97e4T2 1 - 59.3iT - 2.97e4T^{2}
37 1+187.T+5.06e4T2 1 + 187.T + 5.06e4T^{2}
41 1−211.iT−6.89e4T2 1 - 211. iT - 6.89e4T^{2}
43 1+160.iT−7.95e4T2 1 + 160. iT - 7.95e4T^{2}
47 1−539.iT−1.03e5T2 1 - 539. iT - 1.03e5T^{2}
53 1+583.iT−1.48e5T2 1 + 583. iT - 1.48e5T^{2}
59 1+236.T+2.05e5T2 1 + 236.T + 2.05e5T^{2}
61 1+438.iT−2.26e5T2 1 + 438. iT - 2.26e5T^{2}
67 1−639.T+3.00e5T2 1 - 639.T + 3.00e5T^{2}
71 1−79.3iT−3.57e5T2 1 - 79.3iT - 3.57e5T^{2}
73 1+40.6iT−3.89e5T2 1 + 40.6iT - 3.89e5T^{2}
79 1−807.T+4.93e5T2 1 - 807.T + 4.93e5T^{2}
83 1−1.39e3T+5.71e5T2 1 - 1.39e3T + 5.71e5T^{2}
89 1+1.05e3iT−7.04e5T2 1 + 1.05e3iT - 7.04e5T^{2}
97 1−1.00e3iT−9.12e5T2 1 - 1.00e3iT - 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

−11.53666799650851135170409518316, −10.87584106206286645340696038962, −9.682589747089382430201442678124, −8.753562279185520140106136995720, −7.59261388393308480572340229970, −6.44712784942574454934731994346, −5.39921920182866777775900762440, −4.81286722136832984503517846636, −3.27692809030279181221359495473, −2.20858176488484170612119259192, 0.865551365594580357964064637576, 2.32694769031476200239141208591, 3.32471884527527495618411211583, 5.09917030626934351509719817184, 5.61956966384707531022248071384, 6.90032764023071160689080763960, 7.71951610740931399701460647240, 9.199092010495217900981986228711, 10.40621136120507637093999510939, 10.84159398357218634775754806856

Graph of the ZZ-function along the critical line