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Properties

Label 2-3960-5.4-c1-0-41
Degree 22
Conductor 39603960
Sign 0.749+0.662i0.749 + 0.662i
Analytic cond. 31.620731.6207
Root an. cond. 5.623235.62323
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  + (−1.48 + 1.67i)5-s − 0.806i·7-s + 11-s − 2.15i·13-s + 2.54i·17-s + 0.387·19-s + (−0.612 − 4.96i)25-s − 0.649·29-s − 4.96·31-s + (1.35 + 1.19i)35-s − 8.31i·37-s + 2.57·41-s + 0.806i·43-s − 1.61i·47-s + 6.35·49-s + ⋯
L(s)  = 1  + (−0.662 + 0.749i)5-s − 0.304i·7-s + 0.301·11-s − 0.598i·13-s + 0.617i·17-s + 0.0889·19-s + (−0.122 − 0.992i)25-s − 0.120·29-s − 0.891·31-s + (0.228 + 0.201i)35-s − 1.36i·37-s + 0.402·41-s + 0.122i·43-s − 0.235i·47-s + 0.907·49-s + ⋯

Functional equation

Λ(s)=(3960s/2 ΓC(s) L(s)=((0.749+0.662i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3960s/2 ΓC(s+1/2) L(s)=((0.749+0.662i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 39603960    =    23⋅32⋅5⋅112^{3} \cdot 3^{2} \cdot 5 \cdot 11
Sign: 0.749+0.662i0.749 + 0.662i
Analytic conductor: 31.620731.6207
Root analytic conductor: 5.623235.62323
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3960(3169,⋅)\chi_{3960} (3169, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3960, ( :1/2), 0.749+0.662i)(2,\ 3960,\ (\ :1/2),\ 0.749 + 0.662i)

Particular Values

L(1)L(1) ≈\approx 1.3287801121.328780112
L(12)L(\frac12) ≈\approx 1.3287801121.328780112
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+(1.48−1.67i)T 1 + (1.48 - 1.67i)T
11 1−T 1 - T
good7 1+0.806iT−7T2 1 + 0.806iT - 7T^{2}
13 1+2.15iT−13T2 1 + 2.15iT - 13T^{2}
17 1−2.54iT−17T2 1 - 2.54iT - 17T^{2}
19 1−0.387T+19T2 1 - 0.387T + 19T^{2}
23 1−23T2 1 - 23T^{2}
29 1+0.649T+29T2 1 + 0.649T + 29T^{2}
31 1+4.96T+31T2 1 + 4.96T + 31T^{2}
37 1+8.31iT−37T2 1 + 8.31iT - 37T^{2}
41 1−2.57T+41T2 1 - 2.57T + 41T^{2}
43 1−0.806iT−43T2 1 - 0.806iT - 43T^{2}
47 1+1.61iT−47T2 1 + 1.61iT - 47T^{2}
53 1+0.649iT−53T2 1 + 0.649iT - 53T^{2}
59 1+0.649T+59T2 1 + 0.649T + 59T^{2}
61 1+2T+61T2 1 + 2T + 61T^{2}
67 1−3.22iT−67T2 1 - 3.22iT - 67T^{2}
71 1−4.64T+71T2 1 - 4.64T + 71T^{2}
73 1−0.231iT−73T2 1 - 0.231iT - 73T^{2}
79 1+5.53T+79T2 1 + 5.53T + 79T^{2}
83 1+4.08iT−83T2 1 + 4.08iT - 83T^{2}
89 1−4.88T+89T2 1 - 4.88T + 89T^{2}
97 1+13.9iT−97T2 1 + 13.9iT - 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

−8.300619869525827167422648735934, −7.49360812902968019601627651602, −7.12025757518638308247896285041, −6.16950303703722986683957659284, −5.52131563365679899323424929709, −4.36755713742361140147646805301, −3.75182817350942971734839457904, −2.99279226181585655969390146757, −1.91487941403682733760385099362, −0.48678756386793239930030384512, 0.908211194320517475548899421632, 2.00936543981977051858977189747, 3.17528270631638288823377506234, 4.02665370140555710969453826940, 4.75780066247846898622091394497, 5.44391679698015372271050570936, 6.36337637072957191409205898995, 7.18587430024700143159514257431, 7.81830974237169641753753870784, 8.629218900745324072506938948162

Graph of the ZZ-function along the critical line