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Properties

Label 2-976-16.13-c1-0-10
Degree $2$
Conductor $976$
Sign $0.204 + 0.978i$
Analytic cond. $7.79339$
Root an. cond. $2.79166$
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  + (0.211 + 1.39i)2-s + (0.0204 + 0.0204i)3-s + (−1.91 + 0.590i)4-s + (−2.96 + 2.96i)5-s + (−0.0242 + 0.0329i)6-s + 5.25i·7-s + (−1.22 − 2.54i)8-s − 2.99i·9-s + (−4.76 − 3.51i)10-s + (−2.97 + 2.97i)11-s + (−0.0511 − 0.0269i)12-s + (−3.28 − 3.28i)13-s + (−7.35 + 1.10i)14-s − 0.121·15-s + (3.30 − 2.25i)16-s + 3.49·17-s + ⋯
L(s)  = 1  + (0.149 + 0.988i)2-s + (0.0118 + 0.0118i)3-s + (−0.955 + 0.295i)4-s + (−1.32 + 1.32i)5-s + (−0.00990 + 0.0134i)6-s + 1.98i·7-s + (−0.434 − 0.900i)8-s − 0.999i·9-s + (−1.50 − 1.11i)10-s + (−0.898 + 0.898i)11-s + (−0.0147 − 0.00779i)12-s + (−0.910 − 0.910i)13-s + (−1.96 + 0.296i)14-s − 0.0312·15-s + (0.825 − 0.563i)16-s + 0.848·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(976\)    =    \(2^{4} \cdot 61\)
Sign: $0.204 + 0.978i$
Analytic conductor: \(7.79339\)
Root analytic conductor: \(2.79166\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{976} (733, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 976,\ (\ :1/2),\ 0.204 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.381288 - 0.309717i\)
\(L(\frac12)\) \(\approx\) \(0.381288 - 0.309717i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.211 - 1.39i)T \)
61 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (-0.0204 - 0.0204i)T + 3iT^{2} \)
5 \( 1 + (2.96 - 2.96i)T - 5iT^{2} \)
7 \( 1 - 5.25iT - 7T^{2} \)
11 \( 1 + (2.97 - 2.97i)T - 11iT^{2} \)
13 \( 1 + (3.28 + 3.28i)T + 13iT^{2} \)
17 \( 1 - 3.49T + 17T^{2} \)
19 \( 1 + (-3.56 - 3.56i)T + 19iT^{2} \)
23 \( 1 - 1.42iT - 23T^{2} \)
29 \( 1 + (-4.69 - 4.69i)T + 29iT^{2} \)
31 \( 1 - 1.70T + 31T^{2} \)
37 \( 1 + (0.491 - 0.491i)T - 37iT^{2} \)
41 \( 1 + 0.303iT - 41T^{2} \)
43 \( 1 + (-2.22 + 2.22i)T - 43iT^{2} \)
47 \( 1 - 3.37T + 47T^{2} \)
53 \( 1 + (7.72 - 7.72i)T - 53iT^{2} \)
59 \( 1 + (4.65 - 4.65i)T - 59iT^{2} \)
67 \( 1 + (2.13 + 2.13i)T + 67iT^{2} \)
71 \( 1 + 9.36iT - 71T^{2} \)
73 \( 1 + 13.9iT - 73T^{2} \)
79 \( 1 + 7.23T + 79T^{2} \)
83 \( 1 + (6.23 + 6.23i)T + 83iT^{2} \)
89 \( 1 - 0.597iT - 89T^{2} \)
97 \( 1 + 0.337T + 97T^{2} \)
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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

−10.42339501407637950950910121925, −9.730628965255053910268633986919, −8.787554892942739965885641949296, −7.73598293906152666089213030765, −7.56289218913310171241523362984, −6.40972753982390675441848014926, −5.63750584543607098850311790390, −4.74715733886968975291968894161, −3.27983538465739963117406481572, −2.88081080686345220294231652459, 0.25394594630865862038494958913, 1.16431581641509282637434598717, 2.97622876276257384501829069414, 4.09323892347849610119277933365, 4.62192926612553279864362095376, 5.26847720754660730896909639291, 7.20043629670043338327479635798, 7.910892254239545740691545945201, 8.368931308659064064919221045213, 9.598231544506651016446842045573

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