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Properties

Label 2-1984-31.30-c0-0-0
Degree $2$
Conductor $1984$
Sign $1$
Analytic cond. $0.990144$
Root an. cond. $0.995060$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 7-s + 9-s + 19-s + 31-s − 35-s − 41-s + 45-s + 2·47-s + 59-s − 63-s − 2·67-s − 71-s + 81-s + 95-s − 97-s + 101-s − 103-s + 107-s + 109-s − 113-s + ⋯
L(s)  = 1  + 5-s − 7-s + 9-s + 19-s + 31-s − 35-s − 41-s + 45-s + 2·47-s + 59-s − 63-s − 2·67-s − 71-s + 81-s + 95-s − 97-s + 101-s − 103-s + 107-s + 109-s − 113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1984\)    =    \(2^{6} \cdot 31\)
Sign: $1$
Analytic conductor: \(0.990144\)
Root analytic conductor: \(0.995060\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1984} (1921, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1984,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.363970609\)
\(L(\frac12)\) \(\approx\) \(1.363970609\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 - T \)
good3 \( ( 1 - T )( 1 + T ) \)
5 \( 1 - T + T^{2} \)
7 \( 1 + T + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 - T + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( 1 + T + T^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( 1 - T + T^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 + T )^{2} \)
71 \( 1 + T + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( 1 + T + T^{2} \)
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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

−9.498562370271284513168455392276, −8.809220719804173880707610068874, −7.63587780772854693400670068344, −6.93801372505915564084401401634, −6.20032309942401690853112582952, −5.49393077538779529026821463484, −4.47888650250490094258310220839, −3.45949718743552220459799941634, −2.47767351245492657633422028026, −1.29804537939601829628431090955, 1.29804537939601829628431090955, 2.47767351245492657633422028026, 3.45949718743552220459799941634, 4.47888650250490094258310220839, 5.49393077538779529026821463484, 6.20032309942401690853112582952, 6.93801372505915564084401401634, 7.63587780772854693400670068344, 8.809220719804173880707610068874, 9.498562370271284513168455392276

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