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Properties

Label 2-70e2-5.4-c1-0-0
Degree 22
Conductor 49004900
Sign −0.447+0.894i-0.447 + 0.894i
Analytic cond. 39.126639.1266
Root an. cond. 6.255136.25513
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82i·3-s − 5.00·9-s + 4·11-s − 4.24i·13-s + 1.41i·17-s − 2.82·19-s + 4i·23-s − 5.65i·27-s − 8·29-s + 11.3i·33-s − 8i·37-s + 12·39-s − 7.07·41-s + 4i·43-s + 5.65i·47-s + ⋯
L(s)  = 1  + 1.63i·3-s − 1.66·9-s + 1.20·11-s − 1.17i·13-s + 0.342i·17-s − 0.648·19-s + 0.834i·23-s − 1.08i·27-s − 1.48·29-s + 1.96i·33-s − 1.31i·37-s + 1.92·39-s − 1.10·41-s + 0.609i·43-s + 0.825i·47-s + ⋯

Functional equation

Λ(s)=(4900s/2 ΓC(s) L(s)=((−0.447+0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(4900s/2 ΓC(s+1/2) L(s)=((−0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 49004900    =    22⋅52⋅722^{2} \cdot 5^{2} \cdot 7^{2}
Sign: −0.447+0.894i-0.447 + 0.894i
Analytic conductor: 39.126639.1266
Root analytic conductor: 6.255136.25513
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ4900(2549,⋅)\chi_{4900} (2549, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4900, ( :1/2), −0.447+0.894i)(2,\ 4900,\ (\ :1/2),\ -0.447 + 0.894i)

Particular Values

L(1)L(1) ≈\approx 0.31311335480.3131133548
L(12)L(\frac12) ≈\approx 0.31311335480.3131133548
L(32)L(\frac{3}{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
5 1 1
7 1 1
good3 1−2.82iT−3T2 1 - 2.82iT - 3T^{2}
11 1−4T+11T2 1 - 4T + 11T^{2}
13 1+4.24iT−13T2 1 + 4.24iT - 13T^{2}
17 1−1.41iT−17T2 1 - 1.41iT - 17T^{2}
19 1+2.82T+19T2 1 + 2.82T + 19T^{2}
23 1−4iT−23T2 1 - 4iT - 23T^{2}
29 1+8T+29T2 1 + 8T + 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1+8iT−37T2 1 + 8iT - 37T^{2}
41 1+7.07T+41T2 1 + 7.07T + 41T^{2}
43 1−4iT−43T2 1 - 4iT - 43T^{2}
47 1−5.65iT−47T2 1 - 5.65iT - 47T^{2}
53 1+10iT−53T2 1 + 10iT - 53T^{2}
59 1+14.1T+59T2 1 + 14.1T + 59T^{2}
61 1+7.07T+61T2 1 + 7.07T + 61T^{2}
67 1−67T2 1 - 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1−7.07iT−73T2 1 - 7.07iT - 73T^{2}
79 1+8T+79T2 1 + 8T + 79T^{2}
83 1−14.1iT−83T2 1 - 14.1iT - 83T^{2}
89 1+7.07T+89T2 1 + 7.07T + 89T^{2}
97 1+1.41iT−97T2 1 + 1.41iT - 97T^{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

−8.942949144713583328654271604924, −8.220167574700814972990053722298, −7.41185165531857206164805955055, −6.36707529886502573121713057223, −5.65753792160790125380238052392, −5.06976062792617716273536412873, −4.09728979793307579763019489131, −3.72010209563036582183174753313, −2.92175210995622052799884762283, −1.57704306069223156705182802513, 0.080210690097360517716299427673, 1.44360355215686755567363126337, 1.87788481119226131117908204868, 2.93852520467237842583407740284, 3.99692838396465194262923819338, 4.81463451341051355008519600060, 6.04994341538371532034286777114, 6.39026765219186827450966096276, 7.07249938614242741938636024113, 7.52548814642455141748723577550

Graph of the ZZ-function along the critical line