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Properties

Label 2-920-5.4-c1-0-8
Degree $2$
Conductor $920$
Sign $0.716 - 0.697i$
Analytic cond. $7.34623$
Root an. cond. $2.71039$
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  − 0.189i·3-s + (−1.60 + 1.55i)5-s − 1.65i·7-s + 2.96·9-s + 0.0759·11-s + 1.24i·13-s + (0.295 + 0.303i)15-s + 5.17i·17-s − 0.792·19-s − 0.313·21-s + i·23-s + (0.137 − 4.99i)25-s − 1.12i·27-s + 2.85·29-s + 8.90·31-s + ⋯
L(s)  = 1  − 0.109i·3-s + (−0.716 + 0.697i)5-s − 0.626i·7-s + 0.988·9-s + 0.0228·11-s + 0.344i·13-s + (0.0762 + 0.0783i)15-s + 1.25i·17-s − 0.181·19-s − 0.0684·21-s + 0.208i·23-s + (0.0275 − 0.999i)25-s − 0.217i·27-s + 0.529·29-s + 1.59·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.716 - 0.697i$
Analytic conductor: \(7.34623\)
Root analytic conductor: \(2.71039\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 920,\ (\ :1/2),\ 0.716 - 0.697i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31598 + 0.534496i\)
\(L(\frac12)\) \(\approx\) \(1.31598 + 0.534496i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (1.60 - 1.55i)T \)
23 \( 1 - iT \)
good3 \( 1 + 0.189iT - 3T^{2} \)
7 \( 1 + 1.65iT - 7T^{2} \)
11 \( 1 - 0.0759T + 11T^{2} \)
13 \( 1 - 1.24iT - 13T^{2} \)
17 \( 1 - 5.17iT - 17T^{2} \)
19 \( 1 + 0.792T + 19T^{2} \)
29 \( 1 - 2.85T + 29T^{2} \)
31 \( 1 - 8.90T + 31T^{2} \)
37 \( 1 - 6.92iT - 37T^{2} \)
41 \( 1 - 4.64T + 41T^{2} \)
43 \( 1 - 1.75iT - 43T^{2} \)
47 \( 1 - 10.0iT - 47T^{2} \)
53 \( 1 - 11.1iT - 53T^{2} \)
59 \( 1 - 12.4T + 59T^{2} \)
61 \( 1 + 12.0T + 61T^{2} \)
67 \( 1 + 6.96iT - 67T^{2} \)
71 \( 1 + 7.71T + 71T^{2} \)
73 \( 1 - 7.49iT - 73T^{2} \)
79 \( 1 - 15.6T + 79T^{2} \)
83 \( 1 + 6.68iT - 83T^{2} \)
89 \( 1 - 3.03T + 89T^{2} \)
97 \( 1 + 3.21iT - 97T^{2} \)
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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

−10.41876314839773570180036160506, −9.487433807542516981656674094359, −8.249008792796024909401977200987, −7.67018980807896921341866775074, −6.78494616206669107481433941473, −6.18414026011696355753020783101, −4.50243332805999772335218767517, −4.04247133922987666391978862132, −2.82457717220177974943041229158, −1.27230592228001552523739240306, 0.801691221160038984167912539519, 2.42806420700889008767423847545, 3.75234514589954324400077916582, 4.68189935153193221650481134753, 5.39478450146176889338305389612, 6.67515022185093968944755666840, 7.50723122976902491909397955334, 8.348447416314898519221283062871, 9.115440349393535740547416063823, 9.885148132375380503658325198338

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