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Properties

Label 2-980-28.19-c1-0-47
Degree $2$
Conductor $980$
Sign $0.895 + 0.444i$
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 + i)2-s + (−0.866 − 1.5i)3-s + 2i·4-s + (−0.866 − 0.5i)5-s + (0.633 − 2.36i)6-s + (−2 + 2i)8-s + (−0.366 − 1.36i)10-s + (−0.232 + 0.133i)11-s + (3 − 1.73i)12-s + 0.464i·13-s + 1.73i·15-s − 4·16-s + (5.59 − 3.23i)17-s + (3 − 5.19i)19-s + (1 − 1.73i)20-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (−0.499 − 0.866i)3-s + i·4-s + (−0.387 − 0.223i)5-s + (0.258 − 0.965i)6-s + (−0.707 + 0.707i)8-s + (−0.115 − 0.431i)10-s + (−0.0699 + 0.0403i)11-s + (0.866 − 0.499i)12-s + 0.128i·13-s + 0.447i·15-s − 16-s + (1.35 − 0.783i)17-s + (0.688 − 1.19i)19-s + (0.223 − 0.387i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.895 + 0.444i$
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: \((2,\ 980,\ (\ :1/2),\ 0.895 + 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70839 - 0.400209i\)
\(L(\frac12)\) \(\approx\) \(1.70839 - 0.400209i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
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

−9.765815322806444623588525785441, −8.922840836893431517325567979010, −7.79167187751545197284211249974, −7.34554819253744475565230887403, −6.58922457162465287124665859645, −5.65106631481051369569193800102, −4.91465884322222288446602087857, −3.78474654394630699416233358605, −2.64942467845333596070659597045, −0.78479363794904541542502950317, 1.34833358689077515152482957596, 3.02196959501223101946143137640, 3.78939106092217147048336247857, 4.71042419107900204695443587541, 5.48456175449952175717386426678, 6.26052559587522686585144196787, 7.51260997874693778122406818667, 8.508525754498958008783678005276, 9.824208918499417847055090231369, 10.18168378430228549436093252361

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