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Properties

Label 2-980-28.3-c1-0-8
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  + (−1.36 + 0.366i)2-s + (−0.866 + 1.5i)3-s + (1.73 − i)4-s + (0.866 − 0.5i)5-s + (0.633 − 2.36i)6-s + (−1.99 + 2i)8-s + (−0.999 + i)10-s + (0.232 + 0.133i)11-s + 3.46i·12-s + 0.464i·13-s + 1.73i·15-s + (1.99 − 3.46i)16-s + (−5.59 − 3.23i)17-s + (3 + 5.19i)19-s + (0.999 − 1.73i)20-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (−0.499 + 0.866i)3-s + (0.866 − 0.5i)4-s + (0.387 − 0.223i)5-s + (0.258 − 0.965i)6-s + (−0.707 + 0.707i)8-s + (−0.316 + 0.316i)10-s + (0.0699 + 0.0403i)11-s + 0.999i·12-s + 0.128i·13-s + 0.447i·15-s + (0.499 − 0.866i)16-s + (−1.35 − 0.783i)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: $\chi_{980} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.832 - 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.191856 + 0.634834i\)
\(L(\frac12)\) \(\approx\) \(0.191856 + 0.634834i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.366i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 \)
good3 \( 1 + (0.866 - 1.5i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-0.232 - 0.133i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.464iT - 13T^{2} \)
17 \( 1 + (5.59 + 3.23i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.26 - 0.732i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.92T + 29T^{2} \)
31 \( 1 + (3 - 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.73 - 8.19i)T + (-18.5 + 32.0i)T^{2} \)
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 \( 1 + (8.19 - 4.73i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3 + 1.73i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.46iT - 71T^{2} \)
73 \( 1 + (-11.1 - 6.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (12.6 - 7.33i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.4T + 83T^{2} \)
89 \( 1 + (2.19 - 1.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.3iT - 97T^{2} \)
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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

−10.12697873165664065231221739885, −9.657204066433312006355198821790, −8.840907638287204495295903988034, −7.986058326831127183371621909165, −6.97270096051411218646005253480, −6.11330593041256029351180289992, −5.22302657000868786326008806573, −4.39592293053310877347891282952, −2.83077066813627299073876537832, −1.45146309916543315767081435534, 0.45709373755205410600769902256, 1.76591983301320172417892133429, 2.75266743462925037862548373022, 4.19024912753259935971099945036, 5.76401073656522098765701276320, 6.52641404694746417688795934571, 7.08077722257310531224134643805, 7.956161182169358079746282287086, 8.943804045365889199661010441948, 9.566474429906321900816635607522

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