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Properties

Label 2-60e2-3.2-c0-0-3
Degree 22
Conductor 36003600
Sign 0.577+0.816i0.577 + 0.816i
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  + 7-s − 1.41i·11-s + 13-s − 1.41i·17-s − 19-s − 1.41i·23-s + 1.41i·29-s − 31-s − 43-s − 1.41i·47-s + 1.41i·59-s + 61-s + 67-s − 1.41i·77-s + 1.41i·83-s + ⋯
L(s)  = 1  + 7-s − 1.41i·11-s + 13-s − 1.41i·17-s − 19-s − 1.41i·23-s + 1.41i·29-s − 31-s − 43-s − 1.41i·47-s + 1.41i·59-s + 61-s + 67-s − 1.41i·77-s + 1.41i·83-s + ⋯

Functional equation

Λ(s)=(3600s/2 ΓC(s) L(s)=((0.577+0.816i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3600s/2 ΓC(s) L(s)=((0.577+0.816i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 36003600    =    24⋅32⋅522^{4} \cdot 3^{2} \cdot 5^{2}
Sign: 0.577+0.816i0.577 + 0.816i
Analytic conductor: 1.796631.79663
Root analytic conductor: 1.340381.34038
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3600(1601,⋅)\chi_{3600} (1601, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3600, ( :0), 0.577+0.816i)(2,\ 3600,\ (\ :0),\ 0.577 + 0.816i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4009687451.400968745
L(12)L(\frac12) ≈\approx 1.4009687451.400968745
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−T+T2 1 - T + T^{2}
11 1+1.41iT−T2 1 + 1.41iT - T^{2}
13 1−T+T2 1 - T + T^{2}
17 1+1.41iT−T2 1 + 1.41iT - T^{2}
19 1+T+T2 1 + T + T^{2}
23 1+1.41iT−T2 1 + 1.41iT - 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+T+T2 1 + T + T^{2}
47 1+1.41iT−T2 1 + 1.41iT - 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−T+T2 1 - T + T^{2}
71 1−T2 1 - T^{2}
73 1+T2 1 + T^{2}
79 1+T2 1 + T^{2}
83 1−1.41iT−T2 1 - 1.41iT - T^{2}
89 1−T2 1 - T^{2}
97 1−T+T2 1 - T + 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

−8.638500256103326820405106902297, −8.112920924487967028762584241507, −7.06957455539179760687525647156, −6.44428374035513674183502844617, −5.50673453653664649039742821198, −4.93607296628208658586609165469, −3.94764311166269882958010416617, −3.12468352684651830307925912594, −2.06006397745463030029900780347, −0.854814433917733532252852419903, 1.61245723757089528019772218303, 2.02664624170740135540600264529, 3.58369756359856413017064034223, 4.22718171592799916570217338214, 4.98561557376672697548185602270, 5.90005240869906844789575128064, 6.56126826123472603347650751197, 7.55525502599468323412494881251, 8.055589366567735307324079009058, 8.710564308136195994525248405738

Graph of the ZZ-function along the critical line