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Properties

Label 2-336-7.4-c1-0-2
Degree 22
Conductor 336336
Sign 0.991−0.126i0.991 - 0.126i
Analytic cond. 2.682972.68297
Root an. cond. 1.637971.63797
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  + (0.5 − 0.866i)3-s + (1 + 1.73i)5-s + (2.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + (−1 + 1.73i)11-s + 13-s + 1.99·15-s + (0.5 + 0.866i)19-s + (2 − 1.73i)21-s + (0.500 − 0.866i)25-s − 0.999·27-s + 4·29-s + (4.5 − 7.79i)31-s + (0.999 + 1.73i)33-s + (1.00 + 5.19i)35-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.447 + 0.774i)5-s + (0.944 + 0.327i)7-s + (−0.166 − 0.288i)9-s + (−0.301 + 0.522i)11-s + 0.277·13-s + 0.516·15-s + (0.114 + 0.198i)19-s + (0.436 − 0.377i)21-s + (0.100 − 0.173i)25-s − 0.192·27-s + 0.742·29-s + (0.808 − 1.39i)31-s + (0.174 + 0.301i)33-s + (0.169 + 0.878i)35-s + ⋯

Functional equation

Λ(s)=(336s/2 ΓC(s) L(s)=((0.991−0.126i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(336s/2 ΓC(s+1/2) L(s)=((0.991−0.126i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 336336    =    24⋅3⋅72^{4} \cdot 3 \cdot 7
Sign: 0.991−0.126i0.991 - 0.126i
Analytic conductor: 2.682972.68297
Root analytic conductor: 1.637971.63797
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ336(193,⋅)\chi_{336} (193, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 336, ( :1/2), 0.991−0.126i)(2,\ 336,\ (\ :1/2),\ 0.991 - 0.126i)

Particular Values

L(1)L(1) ≈\approx 1.65947+0.105310i1.65947 + 0.105310i
L(12)L(\frac12) ≈\approx 1.65947+0.105310i1.65947 + 0.105310i
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 1
3 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
7 1+(−2.5−0.866i)T 1 + (-2.5 - 0.866i)T
good5 1+(−1−1.73i)T+(−2.5+4.33i)T2 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2}
11 1+(1−1.73i)T+(−5.5−9.52i)T2 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2}
13 1−T+13T2 1 - T + 13T^{2}
17 1+(−8.5−14.7i)T2 1 + (-8.5 - 14.7i)T^{2}
19 1+(−0.5−0.866i)T+(−9.5+16.4i)T2 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2}
23 1+(−11.5+19.9i)T2 1 + (-11.5 + 19.9i)T^{2}
29 1−4T+29T2 1 - 4T + 29T^{2}
31 1+(−4.5+7.79i)T+(−15.5−26.8i)T2 1 + (-4.5 + 7.79i)T + (-15.5 - 26.8i)T^{2}
37 1+(1.5+2.59i)T+(−18.5+32.0i)T2 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2}
41 1+10T+41T2 1 + 10T + 41T^{2}
43 1+5T+43T2 1 + 5T + 43T^{2}
47 1+(3+5.19i)T+(−23.5+40.7i)T2 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2}
53 1+(6−10.3i)T+(−26.5−45.8i)T2 1 + (6 - 10.3i)T + (-26.5 - 45.8i)T^{2}
59 1+(6−10.3i)T+(−29.5−51.0i)T2 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2}
61 1+(5+8.66i)T+(−30.5+52.8i)T2 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2}
67 1+(2.5−4.33i)T+(−33.5−58.0i)T2 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2}
71 1−6T+71T2 1 - 6T + 71T^{2}
73 1+(−1.5+2.59i)T+(−36.5−63.2i)T2 1 + (-1.5 + 2.59i)T + (-36.5 - 63.2i)T^{2}
79 1+(0.5+0.866i)T+(−39.5+68.4i)T2 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2}
83 1+6T+83T2 1 + 6T + 83T^{2}
89 1+(8+13.8i)T+(−44.5+77.0i)T2 1 + (8 + 13.8i)T + (-44.5 + 77.0i)T^{2}
97 1+6T+97T2 1 + 6T + 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

−11.61939431803824956055912989940, −10.64833204944929744237983947565, −9.807703268306811534641644660209, −8.596179846730826081126212669275, −7.80582835615406115758091385680, −6.80391003450114410908091692358, −5.82661766187358698849374095707, −4.56022737392867558812610682089, −2.91328961523927284058033257501, −1.79967279993630797846841592914, 1.47431717086110302589789240265, 3.22664450895118351570326821253, 4.69813009178533278507597604460, 5.26448461159680183834924613663, 6.69096001391946685916152534534, 8.163797806184969046578832758502, 8.575184208198096266151663157800, 9.683132609076294109829606803332, 10.58122589564776834266931025629, 11.40600450184990641011993417741

Graph of the ZZ-function along the critical line