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Properties

Label 2-976-244.243-c0-0-0
Degree $2$
Conductor $976$
Sign $1$
Analytic cond. $0.487087$
Root an. cond. $0.697916$
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  − 2·5-s + 9-s + 2·13-s + 3·25-s + 2·41-s − 2·45-s − 49-s − 61-s − 4·65-s − 2·73-s + 81-s − 2·97-s + 2·109-s + 2·113-s + 2·117-s + ⋯
L(s)  = 1  − 2·5-s + 9-s + 2·13-s + 3·25-s + 2·41-s − 2·45-s − 49-s − 61-s − 4·65-s − 2·73-s + 81-s − 2·97-s + 2·109-s + 2·113-s + 2·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(976\)    =    \(2^{4} \cdot 61\)
Sign: $1$
Analytic conductor: \(0.487087\)
Root analytic conductor: \(0.697916\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{976} (975, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 976,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8370891235\)
\(L(\frac12)\) \(\approx\) \(0.8370891235\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−10.47747820397117813546761939638, −9.177520634229532886131058934607, −8.407633868946449530039779003683, −7.74009308626494715218967657336, −7.02352483917930162281648325955, −6.05610554696550800600773874011, −4.57009387515415029363646557701, −3.99464109349858774179470784762, −3.21253285680080305735151051932, −1.17005849836774558990159470978, 1.17005849836774558990159470978, 3.21253285680080305735151051932, 3.99464109349858774179470784762, 4.57009387515415029363646557701, 6.05610554696550800600773874011, 7.02352483917930162281648325955, 7.74009308626494715218967657336, 8.407633868946449530039779003683, 9.177520634229532886131058934607, 10.47747820397117813546761939638

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