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Properties

Label 2-980-28.27-c1-0-58
Degree $2$
Conductor $980$
Sign $0.753 + 0.657i$
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  + (0.0982 + 1.41i)2-s + 0.662·3-s + (−1.98 + 0.277i)4-s + i·5-s + (0.0650 + 0.934i)6-s + (−0.585 − 2.76i)8-s − 2.56·9-s + (−1.41 + 0.0982i)10-s − 3.61i·11-s + (−1.31 + 0.183i)12-s − 5.83i·13-s + 0.662i·15-s + (3.84 − 1.09i)16-s − 1.36i·17-s + (−0.251 − 3.61i)18-s − 4.09·19-s + ⋯
L(s)  = 1  + (0.0694 + 0.997i)2-s + 0.382·3-s + (−0.990 + 0.138i)4-s + 0.447i·5-s + (0.0265 + 0.381i)6-s + (−0.207 − 0.978i)8-s − 0.853·9-s + (−0.446 + 0.0310i)10-s − 1.08i·11-s + (−0.378 + 0.0530i)12-s − 1.61i·13-s + 0.171i·15-s + (0.961 − 0.274i)16-s − 0.331i·17-s + (−0.0593 − 0.851i)18-s − 0.939·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.753 + 0.657i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ 0.753 + 0.657i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.864521 - 0.324417i\)
\(L(\frac12)\) \(\approx\) \(0.864521 - 0.324417i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0982 - 1.41i)T \)
5 \( 1 - iT \)
7 \( 1 \)
good3 \( 1 - 0.662T + 3T^{2} \)
11 \( 1 + 3.61iT - 11T^{2} \)
13 \( 1 + 5.83iT - 13T^{2} \)
17 \( 1 + 1.36iT - 17T^{2} \)
19 \( 1 + 4.09T + 19T^{2} \)
23 \( 1 + 3.24iT - 23T^{2} \)
29 \( 1 - 5.19T + 29T^{2} \)
31 \( 1 + 8.86T + 31T^{2} \)
37 \( 1 - 10.7T + 37T^{2} \)
41 \( 1 - 0.832iT - 41T^{2} \)
43 \( 1 - 3.10iT - 43T^{2} \)
47 \( 1 + 6.89T + 47T^{2} \)
53 \( 1 + 7.41T + 53T^{2} \)
59 \( 1 - 7.47T + 59T^{2} \)
61 \( 1 + 1.48iT - 61T^{2} \)
67 \( 1 + 2.53iT - 67T^{2} \)
71 \( 1 + 3.52iT - 71T^{2} \)
73 \( 1 + 5.16iT - 73T^{2} \)
79 \( 1 + 11.3iT - 79T^{2} \)
83 \( 1 + 6.49T + 83T^{2} \)
89 \( 1 + 9.39iT - 89T^{2} \)
97 \( 1 - 0.343iT - 97T^{2} \)
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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

−9.717688034369685823305318171171, −8.721828084573625508373196014596, −8.216436227895960406437928796520, −7.54733335521393722227974163326, −6.29663833204245101820586208191, −5.86373655145194872235082996084, −4.83290669691920308001160929796, −3.50227054285996059179940298766, −2.80629527809289081558857306607, −0.39056176151843133378241456168, 1.66490217532702041744037573131, 2.47998723324538917069583148863, 3.85266811484609782037519384387, 4.50629766151454209041521897525, 5.53093503210503769265152016513, 6.68022048338221195889249805920, 7.903437322899075400436857862035, 8.721529945988737530962508810220, 9.335286087017120218522891039203, 9.948085475115944419044407743682

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