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Properties

Label 2-2320-580.339-c0-0-0
Degree $2$
Conductor $2320$
Sign $0.938 - 0.345i$
Analytic cond. $1.15783$
Root an. cond. $1.07602$
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  + (0.222 + 0.974i)5-s + (0.222 − 0.974i)9-s + (1.52 − 0.347i)13-s + 0.867i·17-s + (−0.900 + 0.433i)25-s + (−0.222 − 0.974i)29-s + (1.52 + 0.347i)37-s − 1.24·41-s + 45-s + (0.222 − 0.974i)49-s + (0.376 + 0.781i)53-s + (1.12 + 1.40i)61-s + (0.678 + 1.40i)65-s + (−0.376 + 0.781i)73-s + (−0.900 − 0.433i)81-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)5-s + (0.222 − 0.974i)9-s + (1.52 − 0.347i)13-s + 0.867i·17-s + (−0.900 + 0.433i)25-s + (−0.222 − 0.974i)29-s + (1.52 + 0.347i)37-s − 1.24·41-s + 45-s + (0.222 − 0.974i)49-s + (0.376 + 0.781i)53-s + (1.12 + 1.40i)61-s + (0.678 + 1.40i)65-s + (−0.376 + 0.781i)73-s + (−0.900 − 0.433i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2320\)    =    \(2^{4} \cdot 5 \cdot 29\)
Sign: $0.938 - 0.345i$
Analytic conductor: \(1.15783\)
Root analytic conductor: \(1.07602\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2320} (2079, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2320,\ (\ :0),\ 0.938 - 0.345i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.343929545\)
\(L(\frac12)\) \(\approx\) \(1.343929545\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.222 - 0.974i)T \)
29 \( 1 + (0.222 + 0.974i)T \)
good3 \( 1 + (-0.222 + 0.974i)T^{2} \)
7 \( 1 + (-0.222 + 0.974i)T^{2} \)
11 \( 1 + (0.900 - 0.433i)T^{2} \)
13 \( 1 + (-1.52 + 0.347i)T + (0.900 - 0.433i)T^{2} \)
17 \( 1 - 0.867iT - T^{2} \)
19 \( 1 + (0.222 + 0.974i)T^{2} \)
23 \( 1 + (0.623 + 0.781i)T^{2} \)
31 \( 1 + (-0.623 + 0.781i)T^{2} \)
37 \( 1 + (-1.52 - 0.347i)T + (0.900 + 0.433i)T^{2} \)
41 \( 1 + 1.24T + T^{2} \)
43 \( 1 + (0.623 + 0.781i)T^{2} \)
47 \( 1 + (-0.900 + 0.433i)T^{2} \)
53 \( 1 + (-0.376 - 0.781i)T + (-0.623 + 0.781i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (-1.12 - 1.40i)T + (-0.222 + 0.974i)T^{2} \)
67 \( 1 + (-0.900 - 0.433i)T^{2} \)
71 \( 1 + (0.900 - 0.433i)T^{2} \)
73 \( 1 + (0.376 - 0.781i)T + (-0.623 - 0.781i)T^{2} \)
79 \( 1 + (0.900 + 0.433i)T^{2} \)
83 \( 1 + (-0.222 - 0.974i)T^{2} \)
89 \( 1 + (-0.400 + 0.193i)T + (0.623 - 0.781i)T^{2} \)
97 \( 1 + (-1.22 - 0.974i)T + (0.222 + 0.974i)T^{2} \)
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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

−9.231835348721227941359901766436, −8.462360127261883241561955583625, −7.70256389641720317861911585030, −6.69725407190828192139042179187, −6.23036380824061498660215701952, −5.58240657392985194761589489921, −4.05876042741650366993923274443, −3.61028172593814311635208879132, −2.57533036828524726480071691739, −1.27911164697826269557516783261, 1.18628256905096157088563498168, 2.16739370127896279672385138360, 3.48638343347271282190708895318, 4.46174726798803484051158018184, 5.13655478460359876515496078999, 5.90481584646956947753238594347, 6.81468680516040087491065304677, 7.77423190256531777049839288021, 8.414709344273566797786192152257, 9.077221753847594640007304823488

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