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Properties

Label 2-240-1.1-c7-0-10
Degree 22
Conductor 240240
Sign 11
Analytic cond. 74.972474.9724
Root an. cond. 8.658668.65866
Motivic weight 77
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 27·3-s + 125·5-s + 1.40e3·7-s + 729·9-s + 4.04e3·11-s − 5.89e3·13-s − 3.37e3·15-s + 3.10e4·17-s + 4.03e4·19-s − 3.80e4·21-s + 7.89e4·23-s + 1.56e4·25-s − 1.96e4·27-s − 1.57e5·29-s − 1.14e5·31-s − 1.09e5·33-s + 1.76e5·35-s − 4.71e5·37-s + 1.59e5·39-s − 4.04e5·41-s + 2.53e5·43-s + 9.11e4·45-s − 4.37e5·47-s + 1.15e6·49-s − 8.37e5·51-s + 3.34e5·53-s + 5.05e5·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1.55·7-s + 1/3·9-s + 0.916·11-s − 0.743·13-s − 0.258·15-s + 1.53·17-s + 1.34·19-s − 0.895·21-s + 1.35·23-s + 1/5·25-s − 0.192·27-s − 1.19·29-s − 0.692·31-s − 0.528·33-s + 0.693·35-s − 1.53·37-s + 0.429·39-s − 0.916·41-s + 0.486·43-s + 0.149·45-s − 0.614·47-s + 1.40·49-s − 0.883·51-s + 0.309·53-s + 0.409·55-s + ⋯

Functional equation

Λ(s)=(240s/2 ΓC(s) L(s)=( Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
Λ(s)=(240s/2 ΓC(s+7/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 240240    =    24⋅3⋅52^{4} \cdot 3 \cdot 5
Sign: 11
Analytic conductor: 74.972474.9724
Root analytic conductor: 8.658668.65866
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 240, ( :7/2), 1)(2,\ 240,\ (\ :7/2),\ 1)

Particular Values

L(4)L(4) ≈\approx 2.7819215632.781921563
L(12)L(\frac12) ≈\approx 2.7819215632.781921563
L(92)L(\frac{9}{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+p3T 1 + p^{3} T
5 1−p3T 1 - p^{3} T
good7 1−1408T+p7T2 1 - 1408 T + p^{7} T^{2}
11 1−4044T+p7T2 1 - 4044 T + p^{7} T^{2}
13 1+5890T+p7T2 1 + 5890 T + p^{7} T^{2}
17 1−31002T+p7T2 1 - 31002 T + p^{7} T^{2}
19 1−40300T+p7T2 1 - 40300 T + p^{7} T^{2}
23 1−78912T+p7T2 1 - 78912 T + p^{7} T^{2}
29 1+157194T+p7T2 1 + 157194 T + p^{7} T^{2}
31 1+3704pT+p7T2 1 + 3704 p T + p^{7} T^{2}
37 1+471994T+p7T2 1 + 471994 T + p^{7} T^{2}
41 1+404310T+p7T2 1 + 404310 T + p^{7} T^{2}
43 1−253852T+p7T2 1 - 253852 T + p^{7} T^{2}
47 1+437688T+p7T2 1 + 437688 T + p^{7} T^{2}
53 1−334926T+p7T2 1 - 334926 T + p^{7} T^{2}
59 1+562596T+p7T2 1 + 562596 T + p^{7} T^{2}
61 1−3246662T+p7T2 1 - 3246662 T + p^{7} T^{2}
67 1+3895148T+p7T2 1 + 3895148 T + p^{7} T^{2}
71 1−2345160T+p7T2 1 - 2345160 T + p^{7} T^{2}
73 1−5726954T+p7T2 1 - 5726954 T + p^{7} T^{2}
79 1−5222008T+p7T2 1 - 5222008 T + p^{7} T^{2}
83 1−2928132T+p7T2 1 - 2928132 T + p^{7} T^{2}
89 1+3160230T+p7T2 1 + 3160230 T + p^{7} T^{2}
97 1+1898686T+p7T2 1 + 1898686 T + p^{7} 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

−11.04348464686370767699141247176, −9.950539247164196122483685681750, −9.044649738406282416908099406162, −7.75736531757074696164805966226, −6.98606894726246529228652518213, −5.41703456680094371975914025638, −5.06717257165342080949224187379, −3.52273784841209624245421040435, −1.78096634798540549634832031793, −0.976652657342193778156975566426, 0.976652657342193778156975566426, 1.78096634798540549634832031793, 3.52273784841209624245421040435, 5.06717257165342080949224187379, 5.41703456680094371975914025638, 6.98606894726246529228652518213, 7.75736531757074696164805966226, 9.044649738406282416908099406162, 9.950539247164196122483685681750, 11.04348464686370767699141247176

Graph of the ZZ-function along the critical line