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Properties

Label 2-980-7.4-c1-0-11
Degree $2$
Conductor $980$
Sign $-0.701 + 0.712i$
Analytic cond. $7.82533$
Root an. cond. $2.79738$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)3-s + (−0.5 − 0.866i)5-s + (−0.499 − 0.866i)9-s − 2·13-s + 1.99·15-s + (−3 + 5.19i)17-s + (−2 − 3.46i)19-s + (−3 − 5.19i)23-s + (−0.499 + 0.866i)25-s − 4.00·27-s + 6·29-s + (−2 + 3.46i)31-s + (−1 − 1.73i)37-s + (2 − 3.46i)39-s − 6·41-s + ⋯
L(s)  = 1  + (−0.577 + 0.999i)3-s + (−0.223 − 0.387i)5-s + (−0.166 − 0.288i)9-s − 0.554·13-s + 0.516·15-s + (−0.727 + 1.26i)17-s + (−0.458 − 0.794i)19-s + (−0.625 − 1.08i)23-s + (−0.0999 + 0.173i)25-s − 0.769·27-s + 1.11·29-s + (−0.359 + 0.622i)31-s + (−0.164 − 0.284i)37-s + (0.320 − 0.554i)39-s − 0.937·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.701 + 0.712i$
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: \(1\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.701 + 0.712i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 1.73i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−10.03965015248128628912106807092, −8.765618259406640218291553476730, −8.346358474554124649413682515239, −7.00697775460501886819900456835, −6.18254081660402112783335453292, −5.01832465489147199907528245066, −4.56146916215330830942690266943, −3.59689222064108361214940628665, −2.05404859426744015397564861568, 0, 1.59507968108844440572611551061, 2.81707654877198533725577674054, 4.12663907617630997213205759291, 5.27782986439097672900272675997, 6.22388277960908917403967252649, 6.96309481843633676754822466877, 7.55063861849093784405400132968, 8.469029290848132533946342079075, 9.609312979669962421036077568725

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