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Properties

Label 2-2320-580.243-c0-0-0
Degree $2$
Conductor $2320$
Sign $0.833 + 0.552i$
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  + (0.974 + 0.222i)5-s + (−0.222 − 0.974i)9-s + (−0.119 − 0.189i)13-s − 1.80i·17-s + (0.900 + 0.433i)25-s + (0.974 + 0.222i)29-s + (−0.347 − 1.52i)37-s + (0.158 + 0.158i)41-s − i·45-s + (−0.974 + 0.222i)49-s + (0.623 + 1.78i)53-s + (0.211 + 1.87i)61-s + (−0.0739 − 0.211i)65-s + (0.376 + 0.781i)73-s + (−0.900 + 0.433i)81-s + ⋯
L(s)  = 1  + (0.974 + 0.222i)5-s + (−0.222 − 0.974i)9-s + (−0.119 − 0.189i)13-s − 1.80i·17-s + (0.900 + 0.433i)25-s + (0.974 + 0.222i)29-s + (−0.347 − 1.52i)37-s + (0.158 + 0.158i)41-s − i·45-s + (−0.974 + 0.222i)49-s + (0.623 + 1.78i)53-s + (0.211 + 1.87i)61-s + (−0.0739 − 0.211i)65-s + (0.376 + 0.781i)73-s + (−0.900 + 0.433i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2320\)    =    \(2^{4} \cdot 5 \cdot 29\)
Sign: $0.833 + 0.552i$
Analytic conductor: \(1.15783\)
Root analytic conductor: \(1.07602\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2320} (1983, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2320,\ (\ :0),\ 0.833 + 0.552i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.381773734\)
\(L(\frac12)\) \(\approx\) \(1.381773734\)
\(L(1)\) 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 \( 1 + (-0.974 - 0.222i)T \)
29 \( 1 + (-0.974 - 0.222i)T \)
good3 \( 1 + (0.222 + 0.974i)T^{2} \)
7 \( 1 + (0.974 - 0.222i)T^{2} \)
11 \( 1 + (0.433 - 0.900i)T^{2} \)
13 \( 1 + (0.119 + 0.189i)T + (-0.433 + 0.900i)T^{2} \)
17 \( 1 + 1.80iT - T^{2} \)
19 \( 1 + (0.974 + 0.222i)T^{2} \)
23 \( 1 + (-0.781 - 0.623i)T^{2} \)
31 \( 1 + (-0.781 + 0.623i)T^{2} \)
37 \( 1 + (0.347 + 1.52i)T + (-0.900 + 0.433i)T^{2} \)
41 \( 1 + (-0.158 - 0.158i)T + iT^{2} \)
43 \( 1 + (-0.623 + 0.781i)T^{2} \)
47 \( 1 + (0.900 + 0.433i)T^{2} \)
53 \( 1 + (-0.623 - 1.78i)T + (-0.781 + 0.623i)T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (-0.211 - 1.87i)T + (-0.974 + 0.222i)T^{2} \)
67 \( 1 + (0.433 + 0.900i)T^{2} \)
71 \( 1 + (-0.900 - 0.433i)T^{2} \)
73 \( 1 + (-0.376 - 0.781i)T + (-0.623 + 0.781i)T^{2} \)
79 \( 1 + (0.433 + 0.900i)T^{2} \)
83 \( 1 + (0.974 + 0.222i)T^{2} \)
89 \( 1 + (-0.559 - 1.59i)T + (-0.781 + 0.623i)T^{2} \)
97 \( 1 + (0.777 + 0.974i)T + (-0.222 + 0.974i)T^{2} \)
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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

−9.256767010784778255577000805377, −8.557645414180073031718809401093, −7.34277637689946458873042490574, −6.83617261331328621469753493756, −5.94353075089158949512531555343, −5.32353207847349761135160418543, −4.33900021730659504518331537372, −3.11554565705152443355776235894, −2.48853997705899974942694883237, −1.02911587843854396424917622425, 1.56406536340793072306094015850, 2.31060697843739983234746931459, 3.48220751748494800270161045866, 4.67404872172843023128626346966, 5.25759919102729838813644080548, 6.21062590916149497090557714506, 6.69995591781026019336729893348, 8.027865115272372169526393091402, 8.348003785044959787802868043901, 9.254897606206087200459630134379

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