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Properties

Label 2-980-140.139-c1-0-98
Degree $2$
Conductor $980$
Sign $-0.588 + 0.808i$
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  + (1.38 − 0.295i)2-s − 1.66i·3-s + (1.82 − 0.817i)4-s + (−2.22 + 0.220i)5-s + (−0.491 − 2.29i)6-s + (2.28 − 1.67i)8-s + 0.236·9-s + (−3.01 + 0.962i)10-s − 4.81i·11-s + (−1.35 − 3.03i)12-s + 2.14·13-s + (0.366 + 3.69i)15-s + (2.66 − 2.98i)16-s − 5.02·17-s + (0.326 − 0.0698i)18-s + 1.36·19-s + ⋯
L(s)  = 1  + (0.977 − 0.209i)2-s − 0.959i·3-s + (0.912 − 0.408i)4-s + (−0.995 + 0.0986i)5-s + (−0.200 − 0.938i)6-s + (0.807 − 0.590i)8-s + 0.0787·9-s + (−0.952 + 0.304i)10-s − 1.45i·11-s + (−0.392 − 0.875i)12-s + 0.594·13-s + (0.0946 + 0.955i)15-s + (0.665 − 0.746i)16-s − 1.21·17-s + (0.0770 − 0.0164i)18-s + 0.312·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.588 + 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.588 + 0.808i$
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: \((2,\ 980,\ (\ :1/2),\ -0.588 + 0.808i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10016 - 2.16219i\)
\(L(\frac12)\) \(\approx\) \(1.10016 - 2.16219i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 + 0.295i)T \)
5 \( 1 + (2.22 - 0.220i)T \)
7 \( 1 \)
good3 \( 1 + 1.66iT - 3T^{2} \)
11 \( 1 + 4.81iT - 11T^{2} \)
13 \( 1 - 2.14T + 13T^{2} \)
17 \( 1 + 5.02T + 17T^{2} \)
19 \( 1 - 1.36T + 19T^{2} \)
23 \( 1 + 5.18T + 23T^{2} \)
29 \( 1 + 6.43T + 29T^{2} \)
31 \( 1 - 4.62T + 31T^{2} \)
37 \( 1 - 9.82iT - 37T^{2} \)
41 \( 1 + 4.71iT - 41T^{2} \)
43 \( 1 + 0.141T + 43T^{2} \)
47 \( 1 + 2.55iT - 47T^{2} \)
53 \( 1 + 4.84iT - 53T^{2} \)
59 \( 1 - 14.1T + 59T^{2} \)
61 \( 1 + 10.1iT - 61T^{2} \)
67 \( 1 - 9.64T + 67T^{2} \)
71 \( 1 - 9.58iT - 71T^{2} \)
73 \( 1 - 1.67T + 73T^{2} \)
79 \( 1 - 11.8iT - 79T^{2} \)
83 \( 1 + 0.811iT - 83T^{2} \)
89 \( 1 - 16.0iT - 89T^{2} \)
97 \( 1 + 1.76T + 97T^{2} \)
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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

−9.942480999667496945428290111568, −8.462651645021202855959487143983, −7.979431529234525053867201693595, −6.87394429436870530795189206374, −6.44434882507246218409984017713, −5.39555956988354582780557726910, −4.15097351485718015513071422028, −3.46535346885381107311563917226, −2.23067182868036723679326509566, −0.825655181589445869522065638230, 2.05024142056754997725837740375, 3.49696458549048418983314751735, 4.30218179851763191247506378564, 4.60464491971378821969996693502, 5.79503385184719709603323742621, 6.98050902497646584582941350981, 7.52054126210171622867196079696, 8.563957804455451967723695319483, 9.565891052752660472356206097250, 10.47017606855606258260373068262

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