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Properties

Label 2-4788-133.132-c1-0-66
Degree 22
Conductor 47884788
Sign −0.902−0.429i-0.902 - 0.429i
Analytic cond. 38.232338.2323
Root an. cond. 6.183236.18323
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  − 4.27i·5-s + (1.13 − 2.38i)7-s + 2.15·11-s − 0.274i·17-s + 4.35i·19-s − 8.71·23-s − 13.2·25-s + (−10.2 − 4.86i)35-s − 10.8·43-s − 10.2i·47-s + (−4.41 − 5.43i)49-s − 9.19i·55-s + 15.1i·61-s − 2.98i·73-s + (2.44 − 5.13i)77-s + ⋯
L(s)  = 1  − 1.91i·5-s + (0.429 − 0.902i)7-s + 0.648·11-s − 0.0666i·17-s + 0.999i·19-s − 1.81·23-s − 2.65·25-s + (−1.72 − 0.821i)35-s − 1.65·43-s − 1.49i·47-s + (−0.630 − 0.776i)49-s − 1.23i·55-s + 1.94i·61-s − 0.349i·73-s + (0.278 − 0.585i)77-s + ⋯

Functional equation

Λ(s)=(4788s/2 ΓC(s) L(s)=((−0.902−0.429i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 - 0.429i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(4788s/2 ΓC(s+1/2) L(s)=((−0.902−0.429i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.902 - 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 47884788    =    22⋅32⋅7⋅192^{2} \cdot 3^{2} \cdot 7 \cdot 19
Sign: −0.902−0.429i-0.902 - 0.429i
Analytic conductor: 38.232338.2323
Root analytic conductor: 6.183236.18323
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ4788(3457,⋅)\chi_{4788} (3457, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4788, ( :1/2), −0.902−0.429i)(2,\ 4788,\ (\ :1/2),\ -0.902 - 0.429i)

Particular Values

L(1)L(1) ≈\approx 0.94006175740.9400617574
L(12)L(\frac12) ≈\approx 0.94006175740.9400617574
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 1
7 1+(−1.13+2.38i)T 1 + (-1.13 + 2.38i)T
19 1−4.35iT 1 - 4.35iT
good5 1+4.27iT−5T2 1 + 4.27iT - 5T^{2}
11 1−2.15T+11T2 1 - 2.15T + 11T^{2}
13 1+13T2 1 + 13T^{2}
17 1+0.274iT−17T2 1 + 0.274iT - 17T^{2}
23 1+8.71T+23T2 1 + 8.71T + 23T^{2}
29 1−29T2 1 - 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1−37T2 1 - 37T^{2}
41 1+41T2 1 + 41T^{2}
43 1+10.8T+43T2 1 + 10.8T + 43T^{2}
47 1+10.2iT−47T2 1 + 10.2iT - 47T^{2}
53 1−53T2 1 - 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1−15.1iT−61T2 1 - 15.1iT - 61T^{2}
67 1−67T2 1 - 67T^{2}
71 1−71T2 1 - 71T^{2}
73 1+2.98iT−73T2 1 + 2.98iT - 73T^{2}
79 1−79T2 1 - 79T^{2}
83 1+16iT−83T2 1 + 16iT - 83T^{2}
89 1+89T2 1 + 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

−8.074956729001351382337881056332, −7.34154039710492284458297199739, −6.29771277070161195996649676662, −5.56845994320938185359888370407, −4.84198793863636285278349141516, −4.11869720376379473407117227168, −3.69681731720600744952169197880, −1.88940521856649130362146197349, −1.34031154422252612277520027858, −0.24376123539604450909640035407, 1.80098587634226705777557698142, 2.49839953159653315202832061915, 3.26622630577259000769840519619, 4.06119571803000183462266258181, 5.09206850597463874524555034469, 6.12876850460984176344053991510, 6.38835848113723807933510196080, 7.19070632970462574845336615407, 7.921085743830061977397748803055, 8.563225563598359289180044552178

Graph of the ZZ-function along the critical line