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Properties

Label 2-980-28.19-c1-0-69
Degree $2$
Conductor $980$
Sign $-0.832 + 0.553i$
Analytic cond. $7.82533$
Root an. cond. $2.79738$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.366 − 1.36i)2-s + (0.866 + 1.5i)3-s + (−1.73 − i)4-s + (−0.866 − 0.5i)5-s + (2.36 − 0.633i)6-s + (−2 + 1.99i)8-s + (−1 + 0.999i)10-s + (−3.23 + 1.86i)11-s − 3.46i·12-s − 6.46i·13-s − 1.73i·15-s + (1.99 + 3.46i)16-s + (−0.401 + 0.232i)17-s + (3 − 5.19i)19-s + (0.999 + 1.73i)20-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (0.499 + 0.866i)3-s + (−0.866 − 0.5i)4-s + (−0.387 − 0.223i)5-s + (0.965 − 0.258i)6-s + (−0.707 + 0.707i)8-s + (−0.316 + 0.316i)10-s + (−0.974 + 0.562i)11-s − 0.999i·12-s − 1.79i·13-s − 0.447i·15-s + (0.499 + 0.866i)16-s + (−0.0974 + 0.0562i)17-s + (0.688 − 1.19i)19-s + (0.223 + 0.387i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.832 + 0.553i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (411, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.832 + 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.336191 - 1.11242i\)
\(L(\frac12)\) \(\approx\) \(0.336191 - 1.11242i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 + 1.36i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 \)
good3 \( 1 + (-0.866 - 1.5i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (3.23 - 1.86i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 6.46iT - 13T^{2} \)
17 \( 1 + (0.401 - 0.232i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 + 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.73 + 2.73i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.92T + 29T^{2} \)
31 \( 1 + (3 + 5.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.26 + 2.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.46iT - 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 + (0.866 - 1.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1 + 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.73 - 3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.19 - 1.26i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3 - 1.73i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.535iT - 71T^{2} \)
73 \( 1 + (-0.803 + 0.464i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.30 + 1.33i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.53T + 83T^{2} \)
89 \( 1 + (-8.19 - 4.73i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.39iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.835209104012062573514797716928, −9.104444884980846195014945311608, −8.231397348736337000982812403892, −7.43274672823202712288624210191, −5.73807839247314830796251475371, −5.00945329474286321848250730885, −4.13208922676131467796924101573, −3.26713065551524464606359264259, −2.40107852698790781059058104959, −0.45600774705847804627292204646, 1.76750044904168271443309917819, 3.19461759302428917755175816028, 4.17610422963880362750228012220, 5.29950741341773007624277166594, 6.26639658761087376868059420057, 7.12036620722824185098653102695, 7.72800753672207722861866135157, 8.301827691416650158985511518162, 9.201418410409392246486073457246, 10.12090633570623736287755839233

Graph of the $Z$-function along the critical line