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Properties

Label 2-4788-133.132-c1-0-10
Degree 22
Conductor 47884788
Sign −0.137−0.990i-0.137 - 0.990i
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  − 1.60i·5-s + (2.41 + 1.08i)7-s − 4.20·11-s − 3.26·13-s − 0.666i·17-s + (2.31 + 3.69i)19-s − 1.74·23-s + 2.41·25-s − 8.97i·29-s − 7.89·31-s + (1.74 − 3.88i)35-s + 8.54i·37-s + 9.71·41-s − 0.242·43-s + 11.6i·47-s + ⋯
L(s)  = 1  − 0.719i·5-s + (0.912 + 0.409i)7-s − 1.26·11-s − 0.906·13-s − 0.161i·17-s + (0.530 + 0.847i)19-s − 0.362·23-s + 0.482·25-s − 1.66i·29-s − 1.41·31-s + (0.294 − 0.656i)35-s + 1.40i·37-s + 1.51·41-s − 0.0370·43-s + 1.69i·47-s + ⋯

Functional equation

Λ(s)=(4788s/2 ΓC(s) L(s)=((−0.137−0.990i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.137 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(4788s/2 ΓC(s+1/2) L(s)=((−0.137−0.990i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.137 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 47884788    =    22⋅32⋅7⋅192^{2} \cdot 3^{2} \cdot 7 \cdot 19
Sign: −0.137−0.990i-0.137 - 0.990i
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), −0.137−0.990i)(2,\ 4788,\ (\ :1/2),\ -0.137 - 0.990i)

Particular Values

L(1)L(1) ≈\approx 1.0087509071.008750907
L(12)L(\frac12) ≈\approx 1.0087509071.008750907
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+(−2.41−1.08i)T 1 + (-2.41 - 1.08i)T
19 1+(−2.31−3.69i)T 1 + (-2.31 - 3.69i)T
good5 1+1.60iT−5T2 1 + 1.60iT - 5T^{2}
11 1+4.20T+11T2 1 + 4.20T + 11T^{2}
13 1+3.26T+13T2 1 + 3.26T + 13T^{2}
17 1+0.666iT−17T2 1 + 0.666iT - 17T^{2}
23 1+1.74T+23T2 1 + 1.74T + 23T^{2}
29 1+8.97iT−29T2 1 + 8.97iT - 29T^{2}
31 1+7.89T+31T2 1 + 7.89T + 31T^{2}
37 1−8.54iT−37T2 1 - 8.54iT - 37T^{2}
41 1−9.71T+41T2 1 - 9.71T + 41T^{2}
43 1+0.242T+43T2 1 + 0.242T + 43T^{2}
47 1−11.6iT−47T2 1 - 11.6iT - 47T^{2}
53 1−3.71iT−53T2 1 - 3.71iT - 53T^{2}
59 1+13.7T+59T2 1 + 13.7T + 59T^{2}
61 1−1.53iT−61T2 1 - 1.53iT - 61T^{2}
67 1+12.0iT−67T2 1 + 12.0iT - 67T^{2}
71 1−3.71iT−71T2 1 - 3.71iT - 71T^{2}
73 1−9.81iT−73T2 1 - 9.81iT - 73T^{2}
79 1−8.54iT−79T2 1 - 8.54iT - 79T^{2}
83 1−4.82iT−83T2 1 - 4.82iT - 83T^{2}
89 1+9.71T+89T2 1 + 9.71T + 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

−8.213698471211672280671057461182, −7.902048941185491608986934770139, −7.34376189604844159437389968799, −6.05400134549458281036573173607, −5.48086899077289184444553471818, −4.82030125438107941548627364347, −4.26076626420212026238828710406, −2.92928093436004919070264704788, −2.20643662098708113930348153061, −1.14924543163332610712497995635, 0.27735285221846201686415877450, 1.78453907022497985214775260015, 2.61696501140074960202277761569, 3.40348961043425144356921876089, 4.47950088170877700300571902586, 5.17265609194887244328247688206, 5.70522284928985427943454732328, 7.05296785294663006649026562116, 7.24774336492750096356523543187, 7.891195071209426133466595044925

Graph of the ZZ-function along the critical line