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Properties

Label 2-980-140.139-c1-0-97
Degree $2$
Conductor $980$
Sign $-0.563 + 0.826i$
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  + (0.576 + 1.29i)2-s − 2.50i·3-s + (−1.33 + 1.48i)4-s + (−0.639 − 2.14i)5-s + (3.23 − 1.44i)6-s + (−2.69 − 0.867i)8-s − 3.25·9-s + (2.39 − 2.06i)10-s − 2.25i·11-s + (3.72 + 3.34i)12-s + 5.96·13-s + (−5.35 + 1.60i)15-s + (−0.430 − 3.97i)16-s − 2.00·17-s + (−1.87 − 4.20i)18-s − 7.81·19-s + ⋯
L(s)  = 1  + (0.407 + 0.913i)2-s − 1.44i·3-s + (−0.667 + 0.744i)4-s + (−0.286 − 0.958i)5-s + (1.31 − 0.588i)6-s + (−0.951 − 0.306i)8-s − 1.08·9-s + (0.758 − 0.651i)10-s − 0.678i·11-s + (1.07 + 0.964i)12-s + 1.65·13-s + (−1.38 + 0.413i)15-s + (−0.107 − 0.994i)16-s − 0.486·17-s + (−0.442 − 0.991i)18-s − 1.79·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.563 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.563 + 0.826i$
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: \((2,\ 980,\ (\ :1/2),\ -0.563 + 0.826i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.513294 - 0.971196i\)
\(L(\frac12)\) \(\approx\) \(0.513294 - 0.971196i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.576 - 1.29i)T \)
5 \( 1 + (0.639 + 2.14i)T \)
7 \( 1 \)
good3 \( 1 + 2.50iT - 3T^{2} \)
11 \( 1 + 2.25iT - 11T^{2} \)
13 \( 1 - 5.96T + 13T^{2} \)
17 \( 1 + 2.00T + 17T^{2} \)
19 \( 1 + 7.81T + 19T^{2} \)
23 \( 1 + 2.99T + 23T^{2} \)
29 \( 1 + 4.87T + 29T^{2} \)
31 \( 1 - 1.49T + 31T^{2} \)
37 \( 1 - 4.78iT - 37T^{2} \)
41 \( 1 + 8.82iT - 41T^{2} \)
43 \( 1 - 1.12T + 43T^{2} \)
47 \( 1 + 9.56iT - 47T^{2} \)
53 \( 1 - 7.06iT - 53T^{2} \)
59 \( 1 + 11.4T + 59T^{2} \)
61 \( 1 - 1.21iT - 61T^{2} \)
67 \( 1 - 1.11T + 67T^{2} \)
71 \( 1 + 8.40iT - 71T^{2} \)
73 \( 1 - 5.88T + 73T^{2} \)
79 \( 1 + 12.1iT - 79T^{2} \)
83 \( 1 - 11.1iT - 83T^{2} \)
89 \( 1 + 4.57iT - 89T^{2} \)
97 \( 1 - 4.62T + 97T^{2} \)
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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

−9.086237290280560096205240342783, −8.486161675275840401314420590718, −8.110022771448578019967839900690, −7.11869203574237884642899007496, −6.19306723121130080332764087099, −5.84054866289690574004259330790, −4.46888379026270986908809972769, −3.58153991434028228025157694986, −1.87621388459183671195776340888, −0.43466425106079064757259365859, 2.04795339327876747635332576480, 3.26912688452491026137182428282, 4.04906614969015281873735539230, 4.51015311967516595580254122778, 5.85737806572877580613530695699, 6.53083254216938357005322450035, 8.111044122814674633205098000043, 8.984597525063105224484870990876, 9.745465282309345429218557722985, 10.48340064459453319848845668038

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