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Properties

Label 2-312-312.155-c1-0-18
Degree 22
Conductor 312312
Sign 0.113−0.993i0.113 - 0.993i
Analytic cond. 2.491332.49133
Root an. cond. 1.578391.57839
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.752 + 1.19i)2-s + 1.73·3-s + (−0.866 − 1.80i)4-s + 1.75i·5-s + (−1.30 + 2.07i)6-s + (2.81 + 0.320i)8-s + 2.99·9-s + (−2.09 − 1.31i)10-s + 1.10·11-s + (−1.49 − 3.12i)12-s + 3.60i·13-s + 3.03i·15-s + (−2.49 + 3.12i)16-s + (−2.25 + 3.59i)18-s + (3.15 − 1.51i)20-s + ⋯
L(s)  = 1  + (−0.532 + 0.846i)2-s + 1.00·3-s + (−0.433 − 0.901i)4-s + 0.783i·5-s + (−0.532 + 0.846i)6-s + (0.993 + 0.113i)8-s + 0.999·9-s + (−0.663 − 0.417i)10-s + 0.332·11-s + (−0.433 − 0.901i)12-s + 0.999i·13-s + 0.783i·15-s + (−0.624 + 0.780i)16-s + (−0.532 + 0.846i)18-s + (0.706 − 0.339i)20-s + ⋯

Functional equation

Λ(s)=(312s/2 ΓC(s) L(s)=((0.113−0.993i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(312s/2 ΓC(s+1/2) L(s)=((0.113−0.993i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 312312    =    23⋅3⋅132^{3} \cdot 3 \cdot 13
Sign: 0.113−0.993i0.113 - 0.993i
Analytic conductor: 2.491332.49133
Root analytic conductor: 1.578391.57839
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ312(155,⋅)\chi_{312} (155, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 312, ( :1/2), 0.113−0.993i)(2,\ 312,\ (\ :1/2),\ 0.113 - 0.993i)

Particular Values

L(1)L(1) ≈\approx 1.02255+0.912477i1.02255 + 0.912477i
L(12)L(\frac12) ≈\approx 1.02255+0.912477i1.02255 + 0.912477i
L(32)L(\frac{3}{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+(0.752−1.19i)T 1 + (0.752 - 1.19i)T
3 1−1.73T 1 - 1.73T
13 1−3.60iT 1 - 3.60iT
good5 1−1.75iT−5T2 1 - 1.75iT - 5T^{2}
7 1+7T2 1 + 7T^{2}
11 1−1.10T+11T2 1 - 1.10T + 11T^{2}
17 1−17T2 1 - 17T^{2}
19 1−19T2 1 - 19T^{2}
23 1+23T2 1 + 23T^{2}
29 1+29T2 1 + 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1+37T2 1 + 37T^{2}
41 1+10.1T+41T2 1 + 10.1T + 41T^{2}
43 1+4T+43T2 1 + 4T + 43T^{2}
47 1+10.0iT−47T2 1 + 10.0iT - 47T^{2}
53 1+53T2 1 + 53T^{2}
59 1−15.3T+59T2 1 - 15.3T + 59T^{2}
61 1+7.21iT−61T2 1 + 7.21iT - 61T^{2}
67 1−67T2 1 - 67T^{2}
71 1+16.1iT−71T2 1 + 16.1iT - 71T^{2}
73 1−73T2 1 - 73T^{2}
79 1+14.4iT−79T2 1 + 14.4iT - 79T^{2}
83 1−13.1T+83T2 1 - 13.1T + 83T^{2}
89 1+18.3T+89T2 1 + 18.3T + 89T^{2}
97 1−97T2 1 - 97T^{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

−11.76150149770154442512910280389, −10.59076256896335526656347082120, −9.790944930264312309751835938411, −8.945057030592704456380734094954, −8.134240031510238869984890398152, −7.01871311485049090742622347547, −6.54973789814329705904702681845, −4.88138888580028716771048268666, −3.57600495375720843995593337361, −1.90734758367735076617752238152, 1.27820662276913007429211099988, 2.78094096438367197950775498920, 3.90178490701435354166451066194, 5.06172305045338605043609454373, 7.00001323758669918081997652120, 8.159472256387745527276538829101, 8.610871585631877856682433919312, 9.597446181139234240452748309724, 10.27432909152856690506540220694, 11.44252687879254138984884601749

Graph of the ZZ-function along the critical line