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Properties

Label 2-980-20.19-c0-0-5
Degree 22
Conductor 980980
Sign 0.707+0.707i0.707 + 0.707i
Analytic cond. 0.4890830.489083
Root an. cond. 0.6993450.699345
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  + i·2-s − 4-s + (−0.707 − 0.707i)5-s − i·8-s − 9-s + (0.707 − 0.707i)10-s − 1.41i·13-s + 16-s − 1.41i·17-s − i·18-s + (0.707 + 0.707i)20-s + 1.00i·25-s + 1.41·26-s + i·32-s + 1.41·34-s + ⋯
L(s)  = 1  + i·2-s − 4-s + (−0.707 − 0.707i)5-s − i·8-s − 9-s + (0.707 − 0.707i)10-s − 1.41i·13-s + 16-s − 1.41i·17-s − i·18-s + (0.707 + 0.707i)20-s + 1.00i·25-s + 1.41·26-s + i·32-s + 1.41·34-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((0.707+0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s) L(s)=((0.707+0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: 0.707+0.707i0.707 + 0.707i
Analytic conductor: 0.4890830.489083
Root analytic conductor: 0.6993450.699345
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ980(99,⋅)\chi_{980} (99, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :0), 0.707+0.707i)(2,\ 980,\ (\ :0),\ 0.707 + 0.707i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.55681043540.5568104354
L(12)L(\frac12) ≈\approx 0.55681043540.5568104354
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−iT 1 - iT
5 1+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
7 1 1
good3 1+T2 1 + T^{2}
11 1−T2 1 - T^{2}
13 1+1.41iT−T2 1 + 1.41iT - T^{2}
17 1+1.41iT−T2 1 + 1.41iT - T^{2}
19 1−T2 1 - T^{2}
23 1+T2 1 + T^{2}
29 1+T2 1 + T^{2}
31 1−T2 1 - T^{2}
37 1+2iT−T2 1 + 2iT - T^{2}
41 1+1.41T+T2 1 + 1.41T + T^{2}
43 1+T2 1 + T^{2}
47 1+T2 1 + T^{2}
53 1−T2 1 - T^{2}
59 1−T2 1 - T^{2}
61 1−1.41T+T2 1 - 1.41T + T^{2}
67 1+T2 1 + T^{2}
71 1−T2 1 - T^{2}
73 1−1.41iT−T2 1 - 1.41iT - T^{2}
79 1−T2 1 - T^{2}
83 1+T2 1 + T^{2}
89 1+1.41T+T2 1 + 1.41T + T^{2}
97 1−1.41iT−T2 1 - 1.41iT - 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

−9.807968636322277987689931640934, −8.983558221277207994782848661729, −8.321375610689721246121577038221, −7.68497850518753325857033312683, −6.84961743495114197441092533626, −5.45404758597821470960295222444, −5.30304970252868633953811365137, −4.01903251288157073759077382382, −2.98530253044838178819832232157, −0.53534013729196280025652651376, 1.84120496562006941454486017563, 3.02755718847270443124899092917, 3.83947385823145028259015544178, 4.74815458656768011716886465042, 6.01430082940369843112638691450, 6.89604458435717238916698407724, 8.240522059169689902813379734440, 8.544680885048340428591286843487, 9.667332529496313533860220420156, 10.46238853933234341838056300457

Graph of the ZZ-function along the critical line