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Properties

Label 2-980-7.4-c1-0-7
Degree $2$
Conductor $980$
Sign $0.900 + 0.435i$
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  + (−0.207 + 0.358i)3-s + (−0.5 − 0.866i)5-s + (1.41 + 2.44i)9-s + (1.91 − 3.31i)11-s − 3.58·13-s + 0.414·15-s + (3.20 − 5.55i)17-s + (1.82 + 3.16i)19-s + (−0.292 − 0.507i)23-s + (−0.499 + 0.866i)25-s − 2.41·27-s + 6.65·29-s + (2.29 − 3.97i)31-s + (0.792 + 1.37i)33-s + (1.70 + 2.95i)37-s + ⋯
L(s)  = 1  + (−0.119 + 0.207i)3-s + (−0.223 − 0.387i)5-s + (0.471 + 0.816i)9-s + (0.577 − 0.999i)11-s − 0.994·13-s + 0.106·15-s + (0.777 − 1.34i)17-s + (0.419 + 0.726i)19-s + (−0.0610 − 0.105i)23-s + (−0.0999 + 0.173i)25-s − 0.464·27-s + 1.23·29-s + (0.411 − 0.713i)31-s + (0.138 + 0.239i)33-s + (0.280 + 0.486i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.900 + 0.435i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ 0.900 + 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47607 - 0.338592i\)
\(L(\frac12)\) \(\approx\) \(1.47607 - 0.338592i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
good3 \( 1 + (0.207 - 0.358i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-1.91 + 3.31i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.58T + 13T^{2} \)
17 \( 1 + (-3.20 + 5.55i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.82 - 3.16i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.292 + 0.507i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.65T + 29T^{2} \)
31 \( 1 + (-2.29 + 3.97i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.70 - 2.95i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.585T + 41T^{2} \)
43 \( 1 - 11.6T + 43T^{2} \)
47 \( 1 + (4.44 + 7.70i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.87 + 3.25i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.70 + 2.95i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.58 - 4.47i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.53 + 9.58i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.48T + 71T^{2} \)
73 \( 1 + (-2.58 + 4.47i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.57 + 11.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8T + 83T^{2} \)
89 \( 1 + (-8.48 - 14.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 15.7T + 97T^{2} \)
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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

−9.887409892270975403282921901536, −9.259549288387357679019211533585, −8.124323449934479861253664092451, −7.60803012317682710980700791902, −6.54865130586743349387444195755, −5.40457195855130201633722106893, −4.78318275564310309682968900108, −3.69273170515887622495671188114, −2.49531079680727329042369027709, −0.876242537426661560987527842652, 1.20689452223422705453446108372, 2.62632548322809982686739200978, 3.84890697726163434164552195334, 4.66103492921898148795367846600, 5.90979189465625866316598992282, 6.82988158071526273027994287343, 7.33129201478935181273469079250, 8.322419730160353576563969012688, 9.469213458062353261866145389079, 9.919845692691696890003606228114

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