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Properties

Label 2-980-28.27-c1-0-61
Degree $2$
Conductor $980$
Sign $0.446 - 0.894i$
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  + (1.17 + 0.784i)2-s + 2.99·3-s + (0.767 + 1.84i)4-s + i·5-s + (3.52 + 2.35i)6-s + (−0.546 + 2.77i)8-s + 5.98·9-s + (−0.784 + 1.17i)10-s − 2.23i·11-s + (2.30 + 5.53i)12-s − 3.17i·13-s + 2.99i·15-s + (−2.82 + 2.83i)16-s − 3.44i·17-s + (7.04 + 4.70i)18-s − 2.05·19-s + ⋯
L(s)  = 1  + (0.831 + 0.555i)2-s + 1.73·3-s + (0.383 + 0.923i)4-s + 0.447i·5-s + (1.43 + 0.960i)6-s + (−0.193 + 0.981i)8-s + 1.99·9-s + (−0.248 + 0.372i)10-s − 0.674i·11-s + (0.664 + 1.59i)12-s − 0.879i·13-s + 0.774i·15-s + (−0.705 + 0.708i)16-s − 0.835i·17-s + (1.66 + 1.10i)18-s − 0.470·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.446 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.446 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.446 - 0.894i$
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: \((2,\ 980,\ (\ :1/2),\ 0.446 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.75951 + 2.32501i\)
\(L(\frac12)\) \(\approx\) \(3.75951 + 2.32501i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.17 - 0.784i)T \)
5 \( 1 - iT \)
7 \( 1 \)
good3 \( 1 - 2.99T + 3T^{2} \)
11 \( 1 + 2.23iT - 11T^{2} \)
13 \( 1 + 3.17iT - 13T^{2} \)
17 \( 1 + 3.44iT - 17T^{2} \)
19 \( 1 + 2.05T + 19T^{2} \)
23 \( 1 - 2.66iT - 23T^{2} \)
29 \( 1 + 7.38T + 29T^{2} \)
31 \( 1 - 4.89T + 31T^{2} \)
37 \( 1 + 11.1T + 37T^{2} \)
41 \( 1 + 1.46iT - 41T^{2} \)
43 \( 1 - 9.95iT - 43T^{2} \)
47 \( 1 + 6.12T + 47T^{2} \)
53 \( 1 - 4.65T + 53T^{2} \)
59 \( 1 - 7.11T + 59T^{2} \)
61 \( 1 - 2.53iT - 61T^{2} \)
67 \( 1 + 0.0527iT - 67T^{2} \)
71 \( 1 - 0.212iT - 71T^{2} \)
73 \( 1 + 14.8iT - 73T^{2} \)
79 \( 1 + 0.461iT - 79T^{2} \)
83 \( 1 - 10.9T + 83T^{2} \)
89 \( 1 + 7.02iT - 89T^{2} \)
97 \( 1 - 0.185iT - 97T^{2} \)
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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

−9.974221190791594698385576227227, −9.014614757677361745339236335175, −8.314438975265574071866214795018, −7.64269914486441582319086563494, −6.97693236464052709904971093416, −5.87328537060739947108711748792, −4.75520015346138953698494004948, −3.52728377581674454598404043356, −3.15231887217468327294781574358, −2.10229250721194179269481334841, 1.71261683136366183297568567854, 2.30459185909738030456326848123, 3.62848167897110850129297304236, 4.14051130784717188158348762433, 5.14674801621780780245756109250, 6.54598300901966492662224188829, 7.30798714723281555662939104967, 8.435262765175801982664951315154, 9.019375259686658539043780201646, 9.853266004484854632008113056555

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