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Properties

Label 2-984-41.40-c1-0-8
Degree $2$
Conductor $984$
Sign $-0.147 - 0.989i$
Analytic cond. $7.85727$
Root an. cond. $2.80308$
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  + i·3-s + 3.54·5-s + 5.04i·7-s − 9-s − 1.25i·11-s + 4.25i·13-s + 3.54i·15-s + 0.236i·17-s + 0.840i·19-s − 5.04·21-s − 7.74·23-s + 7.58·25-s − i·27-s − 10.3i·29-s − 4.69·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.58·5-s + 1.90i·7-s − 0.333·9-s − 0.378i·11-s + 1.18i·13-s + 0.916i·15-s + 0.0574i·17-s + 0.192i·19-s − 1.10·21-s − 1.61·23-s + 1.51·25-s − 0.192i·27-s − 1.92i·29-s − 0.843·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.147 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.147 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(984\)    =    \(2^{3} \cdot 3 \cdot 41\)
Sign: $-0.147 - 0.989i$
Analytic conductor: \(7.85727\)
Root analytic conductor: \(2.80308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{984} (409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 984,\ (\ :1/2),\ -0.147 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27974 + 1.48484i\)
\(L(\frac12)\) \(\approx\) \(1.27974 + 1.48484i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - iT \)
41 \( 1 + (-6.33 + 0.944i)T \)
good5 \( 1 - 3.54T + 5T^{2} \)
7 \( 1 - 5.04iT - 7T^{2} \)
11 \( 1 + 1.25iT - 11T^{2} \)
13 \( 1 - 4.25iT - 13T^{2} \)
17 \( 1 - 0.236iT - 17T^{2} \)
19 \( 1 - 0.840iT - 19T^{2} \)
23 \( 1 + 7.74T + 23T^{2} \)
29 \( 1 + 10.3iT - 29T^{2} \)
31 \( 1 + 4.69T + 31T^{2} \)
37 \( 1 - 5.58T + 37T^{2} \)
43 \( 1 - 7.82T + 43T^{2} \)
47 \( 1 + 2.07iT - 47T^{2} \)
53 \( 1 + 1.29iT - 53T^{2} \)
59 \( 1 - 7.40T + 59T^{2} \)
61 \( 1 + 10.7T + 61T^{2} \)
67 \( 1 + 5.18iT - 67T^{2} \)
71 \( 1 - 6.74iT - 71T^{2} \)
73 \( 1 - 8.41T + 73T^{2} \)
79 \( 1 - 11.1iT - 79T^{2} \)
83 \( 1 - 10.0T + 83T^{2} \)
89 \( 1 - 3.21iT - 89T^{2} \)
97 \( 1 - 3.94iT - 97T^{2} \)
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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

−9.847037379035664888108126451998, −9.457932782784956245253204263735, −8.909123698889684151822993835544, −7.973914429609208561282653473340, −6.21994547086711349152047873889, −6.03141239463936448039198657092, −5.25901459334615437110119600898, −4.06257477300678988992102878249, −2.46859545471333518426984876705, −2.05693856051699383596557936336, 0.910136864532009973443132854673, 1.97556763099070301924959740013, 3.27754685924198528306665982792, 4.52515202642055469924122339678, 5.62363041723832725174608589588, 6.34819965171001229047370220152, 7.29676061896446347336742173052, 7.81331993502962527409913221387, 9.090422032917134740449255216381, 9.936765120326557250012630849128

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