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Properties

Label 2-312-104.77-c3-0-13
Degree 22
Conductor 312312
Sign 0.925+0.379i0.925 + 0.379i
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  + (−0.481 − 2.78i)2-s − 3i·3-s + (−7.53 + 2.68i)4-s − 21.7·5-s + (−8.36 + 1.44i)6-s − 13.0i·7-s + (11.1 + 19.7i)8-s − 9·9-s + (10.4 + 60.5i)10-s − 21.2·11-s + (8.05 + 22.6i)12-s + (29.0 + 36.7i)13-s + (−36.3 + 6.28i)14-s + 65.1i·15-s + (49.5 − 40.4i)16-s − 41.3·17-s + ⋯
L(s)  = 1  + (−0.170 − 0.985i)2-s − 0.577i·3-s + (−0.942 + 0.335i)4-s − 1.94·5-s + (−0.568 + 0.0983i)6-s − 0.704i·7-s + (0.491 + 0.871i)8-s − 0.333·9-s + (0.330 + 1.91i)10-s − 0.583·11-s + (0.193 + 0.543i)12-s + (0.619 + 0.784i)13-s + (−0.694 + 0.120i)14-s + 1.12i·15-s + (0.774 − 0.632i)16-s − 0.589·17-s + ⋯

Functional equation

Λ(s)=(312s/2 ΓC(s) L(s)=((0.925+0.379i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(312s/2 ΓC(s+3/2) L(s)=((0.925+0.379i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 312312    =    23⋅3⋅132^{3} \cdot 3 \cdot 13
Sign: 0.925+0.379i0.925 + 0.379i
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), 0.925+0.379i)(2,\ 312,\ (\ :3/2),\ 0.925 + 0.379i)

Particular Values

L(2)L(2) ≈\approx 0.43970229810.4397022981
L(12)L(\frac12) ≈\approx 0.43970229810.4397022981
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+(0.481+2.78i)T 1 + (0.481 + 2.78i)T
3 1+3iT 1 + 3iT
13 1+(−29.0−36.7i)T 1 + (-29.0 - 36.7i)T
good5 1+21.7T+125T2 1 + 21.7T + 125T^{2}
7 1+13.0iT−343T2 1 + 13.0iT - 343T^{2}
11 1+21.2T+1.33e3T2 1 + 21.2T + 1.33e3T^{2}
17 1+41.3T+4.91e3T2 1 + 41.3T + 4.91e3T^{2}
19 1+147.T+6.85e3T2 1 + 147.T + 6.85e3T^{2}
23 1−58.4T+1.21e4T2 1 - 58.4T + 1.21e4T^{2}
29 1+180.iT−2.43e4T2 1 + 180. iT - 2.43e4T^{2}
31 1+272.iT−2.97e4T2 1 + 272. iT - 2.97e4T^{2}
37 1−337.T+5.06e4T2 1 - 337.T + 5.06e4T^{2}
41 1−118.iT−6.89e4T2 1 - 118. iT - 6.89e4T^{2}
43 1−404.iT−7.95e4T2 1 - 404. iT - 7.95e4T^{2}
47 1+145.iT−1.03e5T2 1 + 145. iT - 1.03e5T^{2}
53 1−409.iT−1.48e5T2 1 - 409. iT - 1.48e5T^{2}
59 1+809.T+2.05e5T2 1 + 809.T + 2.05e5T^{2}
61 1+34.5iT−2.26e5T2 1 + 34.5iT - 2.26e5T^{2}
67 1−750.T+3.00e5T2 1 - 750.T + 3.00e5T^{2}
71 1−455.iT−3.57e5T2 1 - 455. iT - 3.57e5T^{2}
73 1−130.iT−3.89e5T2 1 - 130. iT - 3.89e5T^{2}
79 1−108.T+4.93e5T2 1 - 108.T + 4.93e5T^{2}
83 1−201.T+5.71e5T2 1 - 201.T + 5.71e5T^{2}
89 1−1.26e3iT−7.04e5T2 1 - 1.26e3iT - 7.04e5T^{2}
97 1−1.42e3iT−9.12e5T2 1 - 1.42e3iT - 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

−11.15334159485919115073263781517, −10.80723493082111981672845883509, −9.243485806986345309994501448607, −8.139768796318296277833415584427, −7.78683524682816828646249687568, −6.51728792368300507648040453162, −4.40423290589202957271020454453, −4.04236740160421471182801446546, −2.56556471827586503618069264899, −0.77145219759340355305041916036, 0.27351847978852472634049637569, 3.26428858097622082011614204305, 4.32516821699397004564654807688, 5.18976501687228516183380951464, 6.53143272936544919251780707498, 7.59758717778727223125400866878, 8.591403941141846203212703619887, 8.769502497267360231764709093122, 10.55367393163664547798018459353, 11.00734510654454756854399269380

Graph of the ZZ-function along the critical line