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Properties

Label 2-980-7.4-c1-0-2
Degree $2$
Conductor $980$
Sign $0.386 - 0.922i$
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  + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)5-s + (1 + 1.73i)9-s + (−3 + 5.19i)11-s − 2·13-s − 0.999·15-s + (−3 + 5.19i)17-s + (4 + 6.92i)19-s + (−1.5 − 2.59i)23-s + (−0.499 + 0.866i)25-s + 5·27-s + 3·29-s + (1 − 1.73i)31-s + (3 + 5.19i)33-s + (−4 − 6.92i)37-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.223 − 0.387i)5-s + (0.333 + 0.577i)9-s + (−0.904 + 1.56i)11-s − 0.554·13-s − 0.258·15-s + (−0.727 + 1.26i)17-s + (0.917 + 1.58i)19-s + (−0.312 − 0.541i)23-s + (−0.0999 + 0.173i)25-s + 0.962·27-s + 0.557·29-s + (0.179 − 0.311i)31-s + (0.522 + 0.904i)33-s + (−0.657 − 1.13i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.386 - 0.922i$
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: \((2,\ 980,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06125 + 0.705925i\)
\(L(\frac12)\) \(\approx\) \(1.06125 + 0.705925i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4 - 6.92i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6 - 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (5 - 8.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3T + 83T^{2} \)
89 \( 1 + (1.5 + 2.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−10.22665326279681432802530989079, −9.364108686856853557880373933773, −8.188250272607580126965554398450, −7.74296214996388938098857911198, −7.03918237554094277966136532677, −5.83499383544825359569944306323, −4.80882369970073147010761300571, −4.06483709625315044635575586511, −2.46747832701378377104507718087, −1.64973143659609205796173223313, 0.56245545339233962096906289444, 2.77635348021519009364105900721, 3.23958708758297670858193642353, 4.57592037970624273610771017081, 5.34470118528749854473689820698, 6.57048773898033294378105196502, 7.27933364803808815766743741765, 8.281360903314058443916661374250, 9.113886758065741616957605798370, 9.750863008896782505080634156278

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