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Properties

Label 2-2320-580.239-c0-0-2
Degree $2$
Conductor $2320$
Sign $0.226 - 0.974i$
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  + (1.75 + 0.846i)3-s + (0.222 + 0.974i)5-s + (−0.781 − 0.376i)7-s + (1.74 + 2.19i)9-s + (−0.433 + 1.90i)15-s + (−1.05 − 1.32i)21-s + (0.433 − 1.90i)23-s + (−0.900 + 0.433i)25-s + (0.781 + 3.42i)27-s + (−0.900 − 0.433i)29-s + (0.193 − 0.846i)35-s − 1.24·41-s + (0.347 − 1.52i)43-s + (−1.74 + 2.19i)45-s + (−0.541 + 0.678i)47-s + ⋯
L(s)  = 1  + (1.75 + 0.846i)3-s + (0.222 + 0.974i)5-s + (−0.781 − 0.376i)7-s + (1.74 + 2.19i)9-s + (−0.433 + 1.90i)15-s + (−1.05 − 1.32i)21-s + (0.433 − 1.90i)23-s + (−0.900 + 0.433i)25-s + (0.781 + 3.42i)27-s + (−0.900 − 0.433i)29-s + (0.193 − 0.846i)35-s − 1.24·41-s + (0.347 − 1.52i)43-s + (−1.74 + 2.19i)45-s + (−0.541 + 0.678i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.226 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.226 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2320\)    =    \(2^{4} \cdot 5 \cdot 29\)
Sign: $0.226 - 0.974i$
Analytic conductor: \(1.15783\)
Root analytic conductor: \(1.07602\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2320} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2320,\ (\ :0),\ 0.226 - 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.078755400\)
\(L(\frac12)\) \(\approx\) \(2.078755400\)
\(L(1)\) 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 \( 1 + (0.900 + 0.433i)T \)
good3 \( 1 + (-1.75 - 0.846i)T + (0.623 + 0.781i)T^{2} \)
7 \( 1 + (0.781 + 0.376i)T + (0.623 + 0.781i)T^{2} \)
11 \( 1 + (0.222 + 0.974i)T^{2} \)
13 \( 1 + (0.222 + 0.974i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (-0.623 + 0.781i)T^{2} \)
23 \( 1 + (-0.433 + 1.90i)T + (-0.900 - 0.433i)T^{2} \)
31 \( 1 + (0.900 - 0.433i)T^{2} \)
37 \( 1 + (0.222 - 0.974i)T^{2} \)
41 \( 1 + 1.24T + T^{2} \)
43 \( 1 + (-0.347 + 1.52i)T + (-0.900 - 0.433i)T^{2} \)
47 \( 1 + (0.541 - 0.678i)T + (-0.222 - 0.974i)T^{2} \)
53 \( 1 + (0.900 - 0.433i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (-1.80 - 0.867i)T + (0.623 + 0.781i)T^{2} \)
67 \( 1 + (-0.222 + 0.974i)T^{2} \)
71 \( 1 + (0.222 + 0.974i)T^{2} \)
73 \( 1 + (0.900 + 0.433i)T^{2} \)
79 \( 1 + (0.222 - 0.974i)T^{2} \)
83 \( 1 + (-1.40 + 0.678i)T + (0.623 - 0.781i)T^{2} \)
89 \( 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2} \)
97 \( 1 + (-0.623 + 0.781i)T^{2} \)
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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

−9.372635033935129906098830092188, −8.678128259870180868608651082504, −7.949046066587398154429994363466, −7.10929390648731498569757676176, −6.55434932761866322294752457794, −5.22063528714409003568273290602, −4.12827093668923632345646450706, −3.55720613720863023948916015080, −2.77758256141876552961259815313, −2.09442953980796103574932144286, 1.30945175201873503891558579993, 2.12582616272476739933106778804, 3.22576818148475464953404473980, 3.75547003643967298911903765274, 5.02680427488514544905490608952, 6.06271479623477986271853995061, 6.91414942428982925845908993551, 7.66364453829863136401297818461, 8.309404182603510500381358863611, 9.002641016278793909118048018871

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