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Properties

Label 2-980-140.139-c1-0-20
Degree $2$
Conductor $980$
Sign $0.639 - 0.768i$
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  + (−1.23 − 0.691i)2-s + 2.08i·3-s + (1.04 + 1.70i)4-s + (−1.52 − 1.63i)5-s + (1.43 − 2.56i)6-s + (−0.107 − 2.82i)8-s − 1.32·9-s + (0.746 + 3.07i)10-s − 0.775i·11-s + (−3.54 + 2.17i)12-s − 4.18·13-s + (3.40 − 3.16i)15-s + (−1.82 + 3.56i)16-s + 4.18·17-s + (1.63 + 0.917i)18-s + 4.88·19-s + ⋯
L(s)  = 1  + (−0.872 − 0.488i)2-s + 1.20i·3-s + (0.521 + 0.853i)4-s + (−0.680 − 0.732i)5-s + (0.587 − 1.04i)6-s + (−0.0381 − 0.999i)8-s − 0.442·9-s + (0.235 + 0.971i)10-s − 0.233i·11-s + (−1.02 + 0.626i)12-s − 1.16·13-s + (0.879 − 0.817i)15-s + (−0.455 + 0.890i)16-s + 1.01·17-s + (0.385 + 0.216i)18-s + 1.12·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.639 - 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.639 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.639 - 0.768i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ 0.639 - 0.768i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.778396 + 0.364839i\)
\(L(\frac12)\) \(\approx\) \(0.778396 + 0.364839i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.23 + 0.691i)T \)
5 \( 1 + (1.52 + 1.63i)T \)
7 \( 1 \)
good3 \( 1 - 2.08iT - 3T^{2} \)
11 \( 1 + 0.775iT - 11T^{2} \)
13 \( 1 + 4.18T + 13T^{2} \)
17 \( 1 - 4.18T + 17T^{2} \)
19 \( 1 - 4.88T + 19T^{2} \)
23 \( 1 - 1.89T + 23T^{2} \)
29 \( 1 - 9.98T + 29T^{2} \)
31 \( 1 + 10.3T + 31T^{2} \)
37 \( 1 + 3.41iT - 37T^{2} \)
41 \( 1 - 3.02iT - 41T^{2} \)
43 \( 1 - 9.19T + 43T^{2} \)
47 \( 1 - 8.27iT - 47T^{2} \)
53 \( 1 - 2.59iT - 53T^{2} \)
59 \( 1 - 4.60T + 59T^{2} \)
61 \( 1 - 3.26iT - 61T^{2} \)
67 \( 1 + 2.27T + 67T^{2} \)
71 \( 1 - 4.41iT - 71T^{2} \)
73 \( 1 + 2.74T + 73T^{2} \)
79 \( 1 - 14.2iT - 79T^{2} \)
83 \( 1 - 2.36iT - 83T^{2} \)
89 \( 1 - 14.3iT - 89T^{2} \)
97 \( 1 - 14.4T + 97T^{2} \)
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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

−9.931502831807826348952979061277, −9.429529811015222276598378187281, −8.760930239403788722098509866910, −7.74392498697089365678360068279, −7.20891184030837451792023332074, −5.53211743757391210719688625420, −4.65332929270050098170085111538, −3.75611299966315023824772072395, −2.86680278776770330941735039967, −1.02973764213504793887813876370, 0.68649920621641743537243262510, 2.06428603323490194911750807258, 3.15113732874447556561479863324, 4.88243736337196727041334571530, 5.96311970455585650573558335750, 6.95710899695950775183180688641, 7.39218999853252863622543156523, 7.81798576815603935107903949290, 8.864705440317755255090892976608, 9.940894811703314517204267956712

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