Properties

Label 2-980-5.4-c1-0-15
Degree $2$
Conductor $980$
Sign $i$
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  − 1.17i·3-s − 2.23i·5-s + 1.62·9-s + 6.62·11-s − 5.64i·13-s − 2.62·15-s + 7.99i·17-s − 5.00·25-s − 5.42i·27-s + 0.623·29-s − 7.77i·33-s − 6.62·39-s − 3.63i·45-s − 12.4i·47-s + 9.37·51-s + ⋯
L(s)  = 1  − 0.677i·3-s − 0.999i·5-s + 0.541·9-s + 1.99·11-s − 1.56i·13-s − 0.677·15-s + 1.93i·17-s − 1.00·25-s − 1.04i·27-s + 0.115·29-s − 1.35i·33-s − 1.06·39-s − 0.541i·45-s − 1.81i·47-s + 1.31·51-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29985 - 1.29985i\)
\(L(\frac12)\) \(\approx\) \(1.29985 - 1.29985i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.23iT \)
7 \( 1 \)
good3 \( 1 + 1.17iT - 3T^{2} \)
11 \( 1 - 6.62T + 11T^{2} \)
13 \( 1 + 5.64iT - 13T^{2} \)
17 \( 1 - 7.99iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 0.623T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 12.4iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 13.4iT - 73T^{2} \)
79 \( 1 + 15.8T + 79T^{2} \)
83 \( 1 + 8.94iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 12.6iT - 97T^{2} \)
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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

−9.797400572354215322609061772179, −8.724248098356151522652424591881, −8.274345815145422732304616723615, −7.28624704887873087093332972635, −6.33451030073734265750581141732, −5.64489316615954586718782898308, −4.34207175843872856157003739856, −3.63267601031804595877084156077, −1.78369571951982602566572462914, −0.995492384960664786365509229933, 1.59282582991826230632506242780, 3.03018330024072510181298466187, 4.07550942877167258480113261564, 4.62364155774651740206829018616, 6.13161152280342265318704519077, 6.90330749413495969910713994286, 7.33979240469291008243626253616, 8.989329318705106512342297596506, 9.402930681903663592093809750893, 10.01130762203300648473145341076

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