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Properties

Label 2-980-140.87-c0-0-2
Degree $2$
Conductor $980$
Sign $0.852 - 0.522i$
Analytic cond. $0.489083$
Root an. cond. $0.699345$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.866 − 0.499i)4-s + (0.991 − 0.130i)5-s + (0.707 − 0.707i)8-s + (0.866 − 0.5i)9-s + (−0.130 + 0.991i)10-s + (−0.541 − 0.541i)13-s + (0.500 + 0.866i)16-s + (−0.478 − 1.78i)17-s + (0.258 + 0.965i)18-s + (−0.923 − 0.382i)20-s + (0.965 − 0.258i)25-s + (0.662 − 0.382i)26-s + 1.41i·29-s + (−0.965 + 0.258i)32-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.866 − 0.499i)4-s + (0.991 − 0.130i)5-s + (0.707 − 0.707i)8-s + (0.866 − 0.5i)9-s + (−0.130 + 0.991i)10-s + (−0.541 − 0.541i)13-s + (0.500 + 0.866i)16-s + (−0.478 − 1.78i)17-s + (0.258 + 0.965i)18-s + (−0.923 − 0.382i)20-s + (0.965 − 0.258i)25-s + (0.662 − 0.382i)26-s + 1.41i·29-s + (−0.965 + 0.258i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.852 - 0.522i$
Analytic conductor: \(0.489083\)
Root analytic conductor: \(0.699345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :0),\ 0.852 - 0.522i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9981658228\)
\(L(\frac12)\) \(\approx\) \(0.9981658228\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 + (-0.991 + 0.130i)T \)
7 \( 1 \)
good3 \( 1 + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
17 \( 1 + (0.478 + 1.78i)T + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.866 - 0.5i)T^{2} \)
41 \( 1 - 1.84iT - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.662 - 0.382i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (1.78 - 0.478i)T + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
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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

−9.949891664474699247962920870696, −9.388872650220915734935085991557, −8.769939347789925202939550993706, −7.51272964762841724241358818753, −6.96181253228218443042518651156, −6.11420735972290896017198723123, −5.13521602870473934138376470490, −4.51979447176763030358620326016, −2.91297639214625736907317958127, −1.25477056353931840363894589746, 1.70160829947482711061508608904, 2.31622087899353581938902121366, 3.82106621500898543813954369183, 4.63703109214376540600668497018, 5.71076465905455647383471438760, 6.78458231173018318236830441067, 7.80446227393410238429417275331, 8.730205956216935972658920142893, 9.509879179157216747179673978778, 10.28561114655730164408322069808

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