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Properties

Label 2-976-244.107-c0-0-0
Degree $2$
Conductor $976$
Sign $0.831 - 0.556i$
Analytic cond. $0.487087$
Root an. cond. $0.697916$
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  + (1.30 + 1.45i)5-s + (0.309 − 0.951i)9-s + (−0.309 + 0.535i)13-s + (−0.244 − 1.14i)17-s + (−0.295 + 2.81i)25-s + (−1.72 + 0.994i)29-s + (1.01 − 1.40i)37-s + (−1.08 − 0.786i)41-s + (1.78 − 0.795i)45-s + (−0.669 + 0.743i)49-s + (1.41 − 0.459i)53-s + (0.104 + 0.994i)61-s + (−1.18 + 0.251i)65-s + (0.669 − 0.743i)73-s + (−0.809 − 0.587i)81-s + ⋯
L(s)  = 1  + (1.30 + 1.45i)5-s + (0.309 − 0.951i)9-s + (−0.309 + 0.535i)13-s + (−0.244 − 1.14i)17-s + (−0.295 + 2.81i)25-s + (−1.72 + 0.994i)29-s + (1.01 − 1.40i)37-s + (−1.08 − 0.786i)41-s + (1.78 − 0.795i)45-s + (−0.669 + 0.743i)49-s + (1.41 − 0.459i)53-s + (0.104 + 0.994i)61-s + (−1.18 + 0.251i)65-s + (0.669 − 0.743i)73-s + (−0.809 − 0.587i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(976\)    =    \(2^{4} \cdot 61\)
Sign: $0.831 - 0.556i$
Analytic conductor: \(0.487087\)
Root analytic conductor: \(0.697916\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{976} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 976,\ (\ :0),\ 0.831 - 0.556i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.236955693\)
\(L(\frac12)\) \(\approx\) \(1.236955693\)
\(L(1)\) 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 \( 1 + (-0.104 - 0.994i)T \)
good3 \( 1 + (-0.309 + 0.951i)T^{2} \)
5 \( 1 + (-1.30 - 1.45i)T + (-0.104 + 0.994i)T^{2} \)
7 \( 1 + (0.669 - 0.743i)T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.244 + 1.14i)T + (-0.913 + 0.406i)T^{2} \)
19 \( 1 + (-0.669 - 0.743i)T^{2} \)
23 \( 1 + (-0.809 - 0.587i)T^{2} \)
29 \( 1 + (1.72 - 0.994i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.978 - 0.207i)T^{2} \)
37 \( 1 + (-1.01 + 1.40i)T + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (1.08 + 0.786i)T + (0.309 + 0.951i)T^{2} \)
43 \( 1 + (0.913 + 0.406i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-1.41 + 0.459i)T + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.978 + 0.207i)T^{2} \)
67 \( 1 + (-0.104 + 0.994i)T^{2} \)
71 \( 1 + (-0.104 - 0.994i)T^{2} \)
73 \( 1 + (-0.669 + 0.743i)T + (-0.104 - 0.994i)T^{2} \)
79 \( 1 + (0.913 + 0.406i)T^{2} \)
83 \( 1 + (0.978 - 0.207i)T^{2} \)
89 \( 1 + (0.478 + 0.658i)T + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (-0.139 + 1.33i)T + (-0.978 - 0.207i)T^{2} \)
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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

−10.19564586114202015133856982857, −9.450676288921735230160965553292, −9.071088801700405135551098505023, −7.27774207817589931103232600432, −7.01366659137812188144279797584, −6.09319804068406760000425757877, −5.31402066324077842800720580361, −3.84029025970150576609866155514, −2.85001095592685263930763141373, −1.86139187466226053897784048645, 1.47711192037507054059372258397, 2.34852542812518532730533671509, 4.12289628882673948136128488059, 5.04961007370173084572221311733, 5.62840779937746424782390282321, 6.53872628162320501534758015857, 7.975325608706812594697849239633, 8.366974628612709076788568179151, 9.473485898681194393036375241421, 9.933850495941570261085087720709

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