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Properties

Label 2-980-140.139-c1-0-19
Degree $2$
Conductor $980$
Sign $-0.772 - 0.634i$
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  + (−0.255 − 1.39i)2-s + 3.18i·3-s + (−1.86 + 0.711i)4-s + (1.80 + 1.31i)5-s + (4.43 − 0.815i)6-s + (1.46 + 2.41i)8-s − 7.15·9-s + (1.36 − 2.85i)10-s + 4.51i·11-s + (−2.26 − 5.95i)12-s − 2.22·13-s + (−4.19 + 5.75i)15-s + (2.98 − 2.66i)16-s − 2.52·17-s + (1.83 + 9.94i)18-s + 5.21·19-s + ⋯
L(s)  = 1  + (−0.180 − 0.983i)2-s + 1.83i·3-s + (−0.934 + 0.355i)4-s + (0.808 + 0.588i)5-s + (1.80 − 0.332i)6-s + (0.519 + 0.854i)8-s − 2.38·9-s + (0.432 − 0.901i)10-s + 1.36i·11-s + (−0.654 − 1.71i)12-s − 0.617·13-s + (−1.08 + 1.48i)15-s + (0.746 − 0.665i)16-s − 0.613·17-s + (0.431 + 2.34i)18-s + 1.19·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 - 0.634i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.772 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.772 - 0.634i$
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: \((2,\ 980,\ (\ :1/2),\ -0.772 - 0.634i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.365235 + 1.02067i\)
\(L(\frac12)\) \(\approx\) \(0.365235 + 1.02067i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.255 + 1.39i)T \)
5 \( 1 + (-1.80 - 1.31i)T \)
7 \( 1 \)
good3 \( 1 - 3.18iT - 3T^{2} \)
11 \( 1 - 4.51iT - 11T^{2} \)
13 \( 1 + 2.22T + 13T^{2} \)
17 \( 1 + 2.52T + 17T^{2} \)
19 \( 1 - 5.21T + 19T^{2} \)
23 \( 1 + 1.71T + 23T^{2} \)
29 \( 1 + 2.31T + 29T^{2} \)
31 \( 1 - 4.62T + 31T^{2} \)
37 \( 1 + 0.336iT - 37T^{2} \)
41 \( 1 + 3.28iT - 41T^{2} \)
43 \( 1 + 6.66T + 43T^{2} \)
47 \( 1 - 1.44iT - 47T^{2} \)
53 \( 1 + 10.0iT - 53T^{2} \)
59 \( 1 - 3.20T + 59T^{2} \)
61 \( 1 + 6.05iT - 61T^{2} \)
67 \( 1 + 11.1T + 67T^{2} \)
71 \( 1 - 9.15iT - 71T^{2} \)
73 \( 1 - 3.24T + 73T^{2} \)
79 \( 1 - 14.2iT - 79T^{2} \)
83 \( 1 - 11.3iT - 83T^{2} \)
89 \( 1 - 15.2iT - 89T^{2} \)
97 \( 1 + 4.49T + 97T^{2} \)
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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

−10.02862597443009280551667127119, −9.806975217189317727167841708321, −9.297636765261983243905970229008, −8.236167688832109010139482244193, −6.97254050467073586850103181796, −5.48703857837386350260505643453, −4.89521121400578764472089657100, −4.02924184458937410091243964702, −3.03874772670188949370859491209, −2.13040192019643674202094700200, 0.53528947054129035915779256045, 1.59451105614399551545037849650, 3.00319059933365154291657929075, 4.86412184340991253084314728387, 5.85351906682085150728597753197, 6.19041018277654459879101738388, 7.16721496021117492440773687690, 7.87983203544855945617906784320, 8.623885199763550222052709348205, 9.207741657735590922543838178959

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