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Properties

Label 2-2240-280.277-c0-0-2
Degree 22
Conductor 22402240
Sign 0.956+0.290i0.956 + 0.290i
Analytic cond. 1.117901.11790
Root an. cond. 1.057311.05731
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  + (1.36 − 0.366i)3-s + (−0.965 − 0.258i)5-s + (0.707 + 0.707i)7-s + (0.866 − 0.5i)9-s + (0.866 + 0.5i)11-s + (−0.707 − 0.707i)13-s − 1.41·15-s + (−0.5 − 0.866i)19-s + (1.22 + 0.707i)21-s + (0.965 + 0.258i)23-s + (0.866 + 0.499i)25-s + 1.41·29-s + (0.707 − 1.22i)31-s + (1.36 + 0.366i)33-s + (−0.500 − 0.866i)35-s + ⋯
L(s)  = 1  + (1.36 − 0.366i)3-s + (−0.965 − 0.258i)5-s + (0.707 + 0.707i)7-s + (0.866 − 0.5i)9-s + (0.866 + 0.5i)11-s + (−0.707 − 0.707i)13-s − 1.41·15-s + (−0.5 − 0.866i)19-s + (1.22 + 0.707i)21-s + (0.965 + 0.258i)23-s + (0.866 + 0.499i)25-s + 1.41·29-s + (0.707 − 1.22i)31-s + (1.36 + 0.366i)33-s + (−0.500 − 0.866i)35-s + ⋯

Functional equation

Λ(s)=(2240s/2 ΓC(s) L(s)=((0.956+0.290i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2240s/2 ΓC(s) L(s)=((0.956+0.290i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 22402240    =    26⋅5⋅72^{6} \cdot 5 \cdot 7
Sign: 0.956+0.290i0.956 + 0.290i
Analytic conductor: 1.117901.11790
Root analytic conductor: 1.057311.05731
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2240(417,⋅)\chi_{2240} (417, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2240, ( :0), 0.956+0.290i)(2,\ 2240,\ (\ :0),\ 0.956 + 0.290i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.7571231811.757123181
L(12)L(\frac12) ≈\approx 1.7571231811.757123181
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
5 1+(0.965+0.258i)T 1 + (0.965 + 0.258i)T
7 1+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
good3 1+(−1.36+0.366i)T+(0.866−0.5i)T2 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2}
11 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
13 1+(0.707+0.707i)T+iT2 1 + (0.707 + 0.707i)T + iT^{2}
17 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
19 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
23 1+(−0.965−0.258i)T+(0.866+0.5i)T2 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2}
29 1−1.41T+T2 1 - 1.41T + T^{2}
31 1+(−0.707+1.22i)T+(−0.5−0.866i)T2 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2}
37 1+(−0.965−0.258i)T+(0.866+0.5i)T2 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2}
41 1+T+T2 1 + T + T^{2}
43 1+iT2 1 + iT^{2}
47 1+(0.258−0.965i)T+(−0.866−0.5i)T2 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2}
53 1+(0.965−0.258i)T+(0.866−0.5i)T2 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2}
59 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
61 1+(1.22−0.707i)T+(0.5−0.866i)T2 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2}
67 1+(0.366+1.36i)T+(−0.866+0.5i)T2 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2}
71 1+1.41T+T2 1 + 1.41T + T^{2}
73 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
79 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
83 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
89 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
97 1+(−1−i)T+iT2 1 + (-1 - i)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

−9.021011894865317524639829284714, −8.359549087364715266246244973858, −7.80498468971331996058573763493, −7.22650384465936819049157639643, −6.23338213289858797788248847714, −4.82170079110954686252092153632, −4.42940273900140448301864617860, −3.13060545303501495221070449148, −2.58920196728580531845538035283, −1.35544173830992074697060118352, 1.42184147963846898914523592432, 2.75630991348065946169831666608, 3.51710310865447464983579726563, 4.28055493696478398158568074210, 4.80402900896928506494614870743, 6.47066987866771911541175107251, 7.10958264520534036100820567510, 7.929286316013831075788731362543, 8.498454607601325677622236863548, 8.960436201754652682336314167385

Graph of the ZZ-function along the critical line