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Properties

Label 2-4140-5.4-c1-0-28
Degree 22
Conductor 41404140
Sign 0.593+0.804i0.593 + 0.804i
Analytic cond. 33.058033.0580
Root an. cond. 5.749615.74961
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  + (−1.79 + 1.32i)5-s − 0.476i·7-s − 3.39·11-s + 2.28i·13-s + 2.61i·17-s − 5.08·19-s − i·23-s + (1.47 − 4.77i)25-s − 3.19·29-s + 3.65·31-s + (0.632 + 0.857i)35-s − 8.44i·37-s − 5.25·41-s − 0.269i·43-s + 8.26i·47-s + ⋯
L(s)  = 1  + (−0.804 + 0.593i)5-s − 0.180i·7-s − 1.02·11-s + 0.634i·13-s + 0.634i·17-s − 1.16·19-s − 0.208i·23-s + (0.295 − 0.955i)25-s − 0.592·29-s + 0.655·31-s + (0.106 + 0.145i)35-s − 1.38i·37-s − 0.820·41-s − 0.0410i·43-s + 1.20i·47-s + ⋯

Functional equation

Λ(s)=(4140s/2 ΓC(s) L(s)=((0.593+0.804i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.593 + 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(4140s/2 ΓC(s+1/2) L(s)=((0.593+0.804i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.593 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 41404140    =    22⋅32⋅5⋅232^{2} \cdot 3^{2} \cdot 5 \cdot 23
Sign: 0.593+0.804i0.593 + 0.804i
Analytic conductor: 33.058033.0580
Root analytic conductor: 5.749615.74961
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ4140(829,⋅)\chi_{4140} (829, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4140, ( :1/2), 0.593+0.804i)(2,\ 4140,\ (\ :1/2),\ 0.593 + 0.804i)

Particular Values

L(1)L(1) ≈\approx 0.80717126310.8071712631
L(12)L(\frac12) ≈\approx 0.80717126310.8071712631
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
5 1+(1.79−1.32i)T 1 + (1.79 - 1.32i)T
23 1+iT 1 + iT
good7 1+0.476iT−7T2 1 + 0.476iT - 7T^{2}
11 1+3.39T+11T2 1 + 3.39T + 11T^{2}
13 1−2.28iT−13T2 1 - 2.28iT - 13T^{2}
17 1−2.61iT−17T2 1 - 2.61iT - 17T^{2}
19 1+5.08T+19T2 1 + 5.08T + 19T^{2}
29 1+3.19T+29T2 1 + 3.19T + 29T^{2}
31 1−3.65T+31T2 1 - 3.65T + 31T^{2}
37 1+8.44iT−37T2 1 + 8.44iT - 37T^{2}
41 1+5.25T+41T2 1 + 5.25T + 41T^{2}
43 1+0.269iT−43T2 1 + 0.269iT - 43T^{2}
47 1−8.26iT−47T2 1 - 8.26iT - 47T^{2}
53 1+7.77iT−53T2 1 + 7.77iT - 53T^{2}
59 1−8.35T+59T2 1 - 8.35T + 59T^{2}
61 1−5.45T+61T2 1 - 5.45T + 61T^{2}
67 1−6.83iT−67T2 1 - 6.83iT - 67T^{2}
71 1+6.28T+71T2 1 + 6.28T + 71T^{2}
73 1+4.90iT−73T2 1 + 4.90iT - 73T^{2}
79 1−9.29T+79T2 1 - 9.29T + 79T^{2}
83 1−6.19iT−83T2 1 - 6.19iT - 83T^{2}
89 1+0.423T+89T2 1 + 0.423T + 89T^{2}
97 1+8.67iT−97T2 1 + 8.67iT - 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

−8.277312840159466886570880246403, −7.57660512165354677481152183396, −6.91509460360579133248427759038, −6.23279504078321777878176508758, −5.32579372541192491807299440589, −4.32256645928829321364529166014, −3.84007525527954927740538645411, −2.78146557675333073287465271705, −1.97514388169613764405132064046, −0.31281668061671267295653460601, 0.78069405463034485309324069745, 2.18940939791622944136944065656, 3.10756195091028731338168955418, 3.98122312856823406848073016917, 4.90024806157015608038666440950, 5.33617440034480285915478652312, 6.33355233737523634511533439592, 7.23657949639143947628427458634, 7.88333836219916053697401974841, 8.458942658409271495516930030075

Graph of the ZZ-function along the critical line