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Properties

Label 2-980-5.4-c1-0-2
Degree $2$
Conductor $980$
Sign $-0.632 - 0.774i$
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  + 1.73i·3-s + (−1.41 − 1.73i)5-s − 11-s + 1.73i·13-s + (2.99 − 2.44i)15-s + 5.19i·17-s − 2.82·19-s + 2.44i·23-s + (−0.999 + 4.89i)25-s + 5.19i·27-s + 7·29-s − 7.07·31-s − 1.73i·33-s + 7.34i·37-s − 2.99·39-s + ⋯
L(s)  = 1  + 0.999i·3-s + (−0.632 − 0.774i)5-s − 0.301·11-s + 0.480i·13-s + (0.774 − 0.632i)15-s + 1.26i·17-s − 0.648·19-s + 0.510i·23-s + (−0.199 + 0.979i)25-s + 1.00i·27-s + 1.29·29-s − 1.27·31-s − 0.301i·33-s + 1.20i·37-s − 0.480·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.632 - 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.632 - 0.774i$
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: \((2,\ 980,\ (\ :1/2),\ -0.632 - 0.774i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.419536 + 0.884169i\)
\(L(\frac12)\) \(\approx\) \(0.419536 + 0.884169i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 + 1.73i)T \)
7 \( 1 \)
good3 \( 1 - 1.73iT - 3T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 - 1.73iT - 13T^{2} \)
17 \( 1 - 5.19iT - 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 - 2.44iT - 23T^{2} \)
29 \( 1 - 7T + 29T^{2} \)
31 \( 1 + 7.07T + 31T^{2} \)
37 \( 1 - 7.34iT - 37T^{2} \)
41 \( 1 + 7.07T + 41T^{2} \)
43 \( 1 + 9.79iT - 43T^{2} \)
47 \( 1 - 12.1iT - 47T^{2} \)
53 \( 1 - 12.2iT - 53T^{2} \)
59 \( 1 + 7.07T + 59T^{2} \)
61 \( 1 - 14.1T + 61T^{2} \)
67 \( 1 + 12.2iT - 67T^{2} \)
71 \( 1 + 10T + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 3T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 5.19iT - 97T^{2} \)
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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

−10.36491174494227399064044545384, −9.418078231249830883339448985963, −8.732085221206297028357087132275, −8.015053077645827115910096070496, −6.97698557531034818077968814631, −5.81398345663655089806799938639, −4.79381452988828101998530296666, −4.21177954198269661181060492219, −3.34906453776074021685784432036, −1.58661233479883260522956626911, 0.45562131095881848179281836825, 2.14361889691587553951370284885, 3.09910867758932953131867298308, 4.30203119775406946387832885858, 5.46956454392757222239026486666, 6.71213006440912943594182825039, 7.02429709409911982384316956666, 7.913781972609144174578170141543, 8.568894742804875439753322715930, 9.866649520626739805001679251060

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