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Properties

Label 2-980-140.139-c1-0-72
Degree $2$
Conductor $980$
Sign $0.987 + 0.155i$
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  + (−0.255 + 1.39i)2-s + 3.18i·3-s + (−1.86 − 0.711i)4-s + (−1.80 + 1.31i)5-s + (−4.43 − 0.815i)6-s + (1.46 − 2.41i)8-s − 7.15·9-s + (−1.36 − 2.85i)10-s − 4.51i·11-s + (2.26 − 5.95i)12-s + 2.22·13-s + (−4.19 − 5.75i)15-s + (2.98 + 2.66i)16-s + 2.52·17-s + (1.83 − 9.94i)18-s − 5.21·19-s + ⋯
L(s)  = 1  + (−0.180 + 0.983i)2-s + 1.83i·3-s + (−0.934 − 0.355i)4-s + (−0.808 + 0.588i)5-s + (−1.80 − 0.332i)6-s + (0.519 − 0.854i)8-s − 2.38·9-s + (−0.432 − 0.901i)10-s − 1.36i·11-s + (0.654 − 1.71i)12-s + 0.617·13-s + (−1.08 − 1.48i)15-s + (0.746 + 0.665i)16-s + 0.613·17-s + (0.431 − 2.34i)18-s − 1.19·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.987 + 0.155i$
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: \((2,\ 980,\ (\ :1/2),\ 0.987 + 0.155i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0585708 - 0.00456803i\)
\(L(\frac12)\) \(\approx\) \(0.0585708 - 0.00456803i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.255 - 1.39i)T \)
5 \( 1 + (1.80 - 1.31i)T \)
7 \( 1 \)
good3 \( 1 - 3.18iT - 3T^{2} \)
11 \( 1 + 4.51iT - 11T^{2} \)
13 \( 1 - 2.22T + 13T^{2} \)
17 \( 1 - 2.52T + 17T^{2} \)
19 \( 1 + 5.21T + 19T^{2} \)
23 \( 1 + 1.71T + 23T^{2} \)
29 \( 1 + 2.31T + 29T^{2} \)
31 \( 1 + 4.62T + 31T^{2} \)
37 \( 1 - 0.336iT - 37T^{2} \)
41 \( 1 + 3.28iT - 41T^{2} \)
43 \( 1 + 6.66T + 43T^{2} \)
47 \( 1 - 1.44iT - 47T^{2} \)
53 \( 1 - 10.0iT - 53T^{2} \)
59 \( 1 + 3.20T + 59T^{2} \)
61 \( 1 + 6.05iT - 61T^{2} \)
67 \( 1 + 11.1T + 67T^{2} \)
71 \( 1 + 9.15iT - 71T^{2} \)
73 \( 1 + 3.24T + 73T^{2} \)
79 \( 1 + 14.2iT - 79T^{2} \)
83 \( 1 - 11.3iT - 83T^{2} \)
89 \( 1 - 15.2iT - 89T^{2} \)
97 \( 1 - 4.49T + 97T^{2} \)
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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

−9.940154763373168229873236718027, −8.944409427162867363776627477989, −8.506593809172169452580460588727, −7.67876197769553985473105604012, −6.30130813880881510649469949723, −5.72746910253823979590998429142, −4.65940329975623690257479929180, −3.81320717641160889489839966853, −3.26757429355933890290551772052, −0.03073425565956456117160118871, 1.36240626114489518638863736639, 2.12803543675313195275432005969, 3.46265081698509650813120038202, 4.58303605249822891869279623967, 5.72318156809775550970109897559, 6.95337969062812357787386524227, 7.64916505608511568100547127166, 8.333375885210607635429211416127, 8.948634328968352387629504835286, 10.10248197086586251374156161763

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