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Properties

Label 2-980-28.19-c1-0-57
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 + (2.36 + 0.633i)6-s + (−2 − 2i)8-s + (1.36 − 0.366i)10-s + (3.23 − 1.86i)11-s + (3 − 1.73i)12-s + 6.46i·13-s + 1.73i·15-s − 4·16-s + (0.401 − 0.232i)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 + (0.965 + 0.258i)6-s + (−0.707 − 0.707i)8-s + (0.431 − 0.115i)10-s + (0.974 − 0.562i)11-s + (0.866 − 0.499i)12-s + 1.79i·13-s + 0.447i·15-s − 16-s + (0.0974 − 0.0562i)17-s + (0.688 − 1.19i)19-s + (0.223 − 0.387i)20-s + ⋯

Functional equation

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

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

\(L(1)\) \(\approx\) \(2.99363 - 0.701292i\)
\(L(\frac12)\) \(\approx\) \(2.99363 - 0.701292i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.23 + 1.86i)T + (5.5 - 9.52i)T^{2} \)
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

−9.634291186906536367469378367003, −9.432294895478072452174661223190, −8.893020171288233342657895703140, −7.10137340107399113139123141366, −6.43624576467280401218122334766, −5.33750582764610805558959040462, −4.33404985111212398305221185175, −3.70893420468842875302527367107, −2.73209782669451640293103672058, −1.41062262833067843663433618494, 1.46404795327645987567583769843, 2.78348681460242904138161376759, 3.74791603911659849409290197292, 5.05531492080197449220421785946, 5.74188212962889143182853403139, 6.78623086134989161344079735069, 7.44765479984803806410907989438, 8.144446793775995113481691232786, 8.894494574690612449102677351075, 9.934597544689104327964049239467

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