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Properties

Label 2-60e2-15.14-c0-0-2
Degree 22
Conductor 36003600
Sign 0.881+0.472i0.881 + 0.472i
Analytic cond. 1.796631.79663
Root an. cond. 1.340381.34038
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·7-s − 1.41i·11-s − i·13-s + 1.41·17-s + 19-s − 1.41·23-s − 1.41i·29-s − 31-s + i·43-s + 1.41·47-s − 1.41i·59-s + 61-s + i·67-s + 1.41·77-s + 1.41·83-s + ⋯
L(s)  = 1  + i·7-s − 1.41i·11-s − i·13-s + 1.41·17-s + 19-s − 1.41·23-s − 1.41i·29-s − 31-s + i·43-s + 1.41·47-s − 1.41i·59-s + 61-s + i·67-s + 1.41·77-s + 1.41·83-s + ⋯

Functional equation

Λ(s)=(3600s/2 ΓC(s) L(s)=((0.881+0.472i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3600s/2 ΓC(s) L(s)=((0.881+0.472i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 36003600    =    24⋅32⋅522^{4} \cdot 3^{2} \cdot 5^{2}
Sign: 0.881+0.472i0.881 + 0.472i
Analytic conductor: 1.796631.79663
Root analytic conductor: 1.340381.34038
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3600(449,⋅)\chi_{3600} (449, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3600, ( :0), 0.881+0.472i)(2,\ 3600,\ (\ :0),\ 0.881 + 0.472i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2964248321.296424832
L(12)L(\frac12) ≈\approx 1.2964248321.296424832
L(1)L(1) 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
good7 1−iT−T2 1 - iT - T^{2}
11 1+1.41iT−T2 1 + 1.41iT - T^{2}
13 1+iT−T2 1 + iT - T^{2}
17 1−1.41T+T2 1 - 1.41T + T^{2}
19 1−T+T2 1 - T + T^{2}
23 1+1.41T+T2 1 + 1.41T + T^{2}
29 1+1.41iT−T2 1 + 1.41iT - T^{2}
31 1+T+T2 1 + T + T^{2}
37 1−T2 1 - T^{2}
41 1−T2 1 - T^{2}
43 1−iT−T2 1 - iT - T^{2}
47 1−1.41T+T2 1 - 1.41T + T^{2}
53 1+T2 1 + T^{2}
59 1+1.41iT−T2 1 + 1.41iT - T^{2}
61 1−T+T2 1 - T + T^{2}
67 1−iT−T2 1 - iT - T^{2}
71 1−T2 1 - T^{2}
73 1−T2 1 - T^{2}
79 1+T2 1 + T^{2}
83 1−1.41T+T2 1 - 1.41T + T^{2}
89 1−T2 1 - T^{2}
97 1−iT−T2 1 - iT - T^{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

−8.491319359496130998365278313813, −8.048926040971543152693827590152, −7.41506294075812564212143112229, −6.01053893254590270805515487532, −5.82436061450194777945724221266, −5.17938271585647593745218092500, −3.80522900517092784242027873356, −3.17960553631215887323214806857, −2.30011315684561916291634291834, −0.870213741523515113660782116175, 1.26325894676897359106920697946, 2.17041713219627770168738164403, 3.54246297667616742686760450820, 4.06676293688038397551329429009, 4.97557605076942760042474953897, 5.71893088329926182108478285573, 6.81990408756708762658555072224, 7.36749747949099579525022890121, 7.73590712605350375653661837162, 8.925234705519217595961433380630

Graph of the ZZ-function along the critical line