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Properties

Label 2-920-5.4-c1-0-29
Degree $2$
Conductor $920$
Sign $-0.946 + 0.321i$
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  − 2.49i·3-s + (2.11 − 0.719i)5-s − 2.92i·7-s − 3.24·9-s − 4.10·11-s − 0.0122i·13-s + (−1.79 − 5.29i)15-s − 0.155i·17-s − 4.32·19-s − 7.31·21-s − i·23-s + (3.96 − 3.04i)25-s + 0.616i·27-s + 6.79·29-s + 2.20·31-s + ⋯
L(s)  = 1  − 1.44i·3-s + (0.946 − 0.321i)5-s − 1.10i·7-s − 1.08·9-s − 1.23·11-s − 0.00341i·13-s + (−0.464 − 1.36i)15-s − 0.0376i·17-s − 0.992·19-s − 1.59·21-s − 0.208i·23-s + (0.792 − 0.609i)25-s + 0.118i·27-s + 1.26·29-s + 0.396·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 + 0.321i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.946 + 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $-0.946 + 0.321i$
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: \((2,\ 920,\ (\ :1/2),\ -0.946 + 0.321i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.242952 - 1.46987i\)
\(L(\frac12)\) \(\approx\) \(0.242952 - 1.46987i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.11 + 0.719i)T \)
23 \( 1 + iT \)
good3 \( 1 + 2.49iT - 3T^{2} \)
7 \( 1 + 2.92iT - 7T^{2} \)
11 \( 1 + 4.10T + 11T^{2} \)
13 \( 1 + 0.0122iT - 13T^{2} \)
17 \( 1 + 0.155iT - 17T^{2} \)
19 \( 1 + 4.32T + 19T^{2} \)
29 \( 1 - 6.79T + 29T^{2} \)
31 \( 1 - 2.20T + 31T^{2} \)
37 \( 1 - 4.60iT - 37T^{2} \)
41 \( 1 + 7.67T + 41T^{2} \)
43 \( 1 - 8.38iT - 43T^{2} \)
47 \( 1 + 6.38iT - 47T^{2} \)
53 \( 1 + 7.80iT - 53T^{2} \)
59 \( 1 - 14.1T + 59T^{2} \)
61 \( 1 - 7.05T + 61T^{2} \)
67 \( 1 + 7.31iT - 67T^{2} \)
71 \( 1 + 5.84T + 71T^{2} \)
73 \( 1 - 0.727iT - 73T^{2} \)
79 \( 1 + 4.81T + 79T^{2} \)
83 \( 1 - 7.75iT - 83T^{2} \)
89 \( 1 + 6.77T + 89T^{2} \)
97 \( 1 + 14.8iT - 97T^{2} \)
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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

−10.00632379203320298267778580088, −8.456248544729180797037192372816, −8.110915260557925140707074486440, −6.93631892298767056475268867787, −6.57690537801993171667937310722, −5.47759447193571755434214134688, −4.48919438583463313716971013628, −2.82841626576254440387751195497, −1.84838442469030958013631985726, −0.67930528228727339301741264084, 2.26024262144316486011324711554, 3.01780341038949710201133707063, 4.35237532473430386316103441843, 5.32108864829326235697813843811, 5.74665906504392104366844955800, 6.89666103412951487395758445099, 8.357914225827067329732550475102, 8.901490181150273268502562211281, 9.784223758567199143403982096155, 10.36635222404574964360594276110

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