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Properties

Label 2-980-5.4-c1-0-5
Degree 22
Conductor 980980
Sign −0.894−0.447i-0.894 - 0.447i
Analytic cond. 7.825337.82533
Root an. cond. 2.797382.79738
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  + 3i·3-s + (2 + i)5-s − 6·9-s + 3·11-s + i·13-s + (−3 + 6i)15-s + 5i·17-s − 8·19-s − 2i·23-s + (3 + 4i)25-s − 9i·27-s + 29-s + 2·31-s + 9i·33-s + 10i·37-s + ⋯
L(s)  = 1  + 1.73i·3-s + (0.894 + 0.447i)5-s − 2·9-s + 0.904·11-s + 0.277i·13-s + (−0.774 + 1.54i)15-s + 1.21i·17-s − 1.83·19-s − 0.417i·23-s + (0.600 + 0.800i)25-s − 1.73i·27-s + 0.185·29-s + 0.359·31-s + 1.56i·33-s + 1.64i·37-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((−0.894−0.447i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((−0.894−0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: −0.894−0.447i-0.894 - 0.447i
Analytic conductor: 7.825337.82533
Root analytic conductor: 2.797382.79738
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ980(589,⋅)\chi_{980} (589, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :1/2), −0.894−0.447i)(2,\ 980,\ (\ :1/2),\ -0.894 - 0.447i)

Particular Values

L(1)L(1) ≈\approx 0.384578+1.62909i0.384578 + 1.62909i
L(12)L(\frac12) ≈\approx 0.384578+1.62909i0.384578 + 1.62909i
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+(−2−i)T 1 + (-2 - i)T
7 1 1
good3 1−3iT−3T2 1 - 3iT - 3T^{2}
11 1−3T+11T2 1 - 3T + 11T^{2}
13 1−iT−13T2 1 - iT - 13T^{2}
17 1−5iT−17T2 1 - 5iT - 17T^{2}
19 1+8T+19T2 1 + 8T + 19T^{2}
23 1+2iT−23T2 1 + 2iT - 23T^{2}
29 1−T+29T2 1 - T + 29T^{2}
31 1−2T+31T2 1 - 2T + 31T^{2}
37 1−10iT−37T2 1 - 10iT - 37T^{2}
41 1−6T+41T2 1 - 6T + 41T^{2}
43 1−4iT−43T2 1 - 4iT - 43T^{2}
47 1+11iT−47T2 1 + 11iT - 47T^{2}
53 1+6iT−53T2 1 + 6iT - 53T^{2}
59 1+10T+59T2 1 + 10T + 59T^{2}
61 1+61T2 1 + 61T^{2}
67 1+10iT−67T2 1 + 10iT - 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1+10iT−73T2 1 + 10iT - 73T^{2}
79 1−7T+79T2 1 - 7T + 79T^{2}
83 1−12iT−83T2 1 - 12iT - 83T^{2}
89 1−8T+89T2 1 - 8T + 89T^{2}
97 1+3iT−97T2 1 + 3iT - 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

−10.41335474582043844198113590174, −9.622560021151432896492148975476, −8.935894304999737881634882744772, −8.261941328595009393414374843364, −6.48267042328027339439777400728, −6.17872031279806468540302374260, −4.92756198997538272549689692150, −4.18923555886163601576632259182, −3.31062373178681191280085470489, −2.01841951785734878764055362974, 0.77182193178669602856374884423, 1.86820668411512967594478598834, 2.71340007281772235735847648629, 4.41093566146988608438378054487, 5.70687099983519711550228008734, 6.27392318827859543239492030933, 7.03991450957754779616423359108, 7.85331663001520225087384247214, 8.839989837474824533469769095291, 9.308647782440550662308877980361

Graph of the ZZ-function along the critical line