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Properties

Label 2-1200-60.23-c0-0-2
Degree 22
Conductor 12001200
Sign 0.923+0.382i0.923 + 0.382i
Analytic cond. 0.5988780.598878
Root an. cond. 0.7738720.773872
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  + (0.707 − 0.707i)3-s + (0.707 + 0.707i)7-s − 1.00i·9-s + (1.22 + 1.22i)13-s − 1.73·19-s + 1.00·21-s + (−0.707 − 0.707i)27-s − 1.73i·31-s + 1.73·39-s + (−0.707 + 0.707i)43-s + (−1.22 + 1.22i)57-s − 61-s + (0.707 − 0.707i)63-s + (−0.707 − 0.707i)67-s − 1.00·81-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s + (0.707 + 0.707i)7-s − 1.00i·9-s + (1.22 + 1.22i)13-s − 1.73·19-s + 1.00·21-s + (−0.707 − 0.707i)27-s − 1.73i·31-s + 1.73·39-s + (−0.707 + 0.707i)43-s + (−1.22 + 1.22i)57-s − 61-s + (0.707 − 0.707i)63-s + (−0.707 − 0.707i)67-s − 1.00·81-s + ⋯

Functional equation

Λ(s)=(1200s/2 ΓC(s) L(s)=((0.923+0.382i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1200s/2 ΓC(s) L(s)=((0.923+0.382i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12001200    =    24⋅3⋅522^{4} \cdot 3 \cdot 5^{2}
Sign: 0.923+0.382i0.923 + 0.382i
Analytic conductor: 0.5988780.598878
Root analytic conductor: 0.7738720.773872
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1200(143,⋅)\chi_{1200} (143, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1200, ( :0), 0.923+0.382i)(2,\ 1200,\ (\ :0),\ 0.923 + 0.382i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4252624021.425262402
L(12)L(\frac12) ≈\approx 1.4252624021.425262402
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+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
5 1 1
good7 1+(−0.707−0.707i)T+iT2 1 + (-0.707 - 0.707i)T + iT^{2}
11 1+T2 1 + T^{2}
13 1+(−1.22−1.22i)T+iT2 1 + (-1.22 - 1.22i)T + iT^{2}
17 1+iT2 1 + iT^{2}
19 1+1.73T+T2 1 + 1.73T + T^{2}
23 1+iT2 1 + iT^{2}
29 1+T2 1 + T^{2}
31 1+1.73iT−T2 1 + 1.73iT - T^{2}
37 1−iT2 1 - iT^{2}
41 1−T2 1 - T^{2}
43 1+(0.707−0.707i)T−iT2 1 + (0.707 - 0.707i)T - iT^{2}
47 1−iT2 1 - iT^{2}
53 1−iT2 1 - iT^{2}
59 1−T2 1 - T^{2}
61 1+T+T2 1 + T + T^{2}
67 1+(0.707+0.707i)T+iT2 1 + (0.707 + 0.707i)T + iT^{2}
71 1+T2 1 + T^{2}
73 1+iT2 1 + iT^{2}
79 1+T2 1 + T^{2}
83 1+iT2 1 + iT^{2}
89 1+T2 1 + T^{2}
97 1+(−1.22+1.22i)T−iT2 1 + (-1.22 + 1.22i)T - iT^{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

−9.616820441958502558689855416077, −8.813374709990459824199219602287, −8.422264989579997865737134093847, −7.58503914190711347193697991637, −6.45724699722851222706444411063, −6.04652514503896561254383975070, −4.57857467231428930620567692770, −3.73483921743346343209832230530, −2.36489843690771134487902723627, −1.63706512570907164309288813831, 1.59523592017145420401827546357, 2.99361070775509986676520892073, 3.89276714175002455734770705118, 4.67662173711173281959677255772, 5.63812588286002796356369480464, 6.76882675492295709098418841521, 7.84596027834239965806247751179, 8.415464569118885580803494365794, 8.976464430712597153367767603591, 10.27697434287880058642552078185

Graph of the ZZ-function along the critical line