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Properties

Label 2-2320-580.3-c0-0-0
Degree $2$
Conductor $2320$
Sign $0.784 - 0.619i$
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  + (−0.781 − 0.623i)5-s + (0.623 + 0.781i)9-s + (0.211 + 1.87i)13-s − 0.445i·17-s + (0.222 + 0.974i)25-s + (−0.781 − 0.623i)29-s + (0.541 + 0.678i)37-s + (1.33 + 1.33i)41-s − 0.999i·45-s + (0.781 − 0.623i)49-s + (−0.900 + 1.43i)53-s + (1.59 − 0.559i)61-s + (1.00 − 1.59i)65-s + (1.90 + 0.433i)73-s + (−0.222 + 0.974i)81-s + ⋯
L(s)  = 1  + (−0.781 − 0.623i)5-s + (0.623 + 0.781i)9-s + (0.211 + 1.87i)13-s − 0.445i·17-s + (0.222 + 0.974i)25-s + (−0.781 − 0.623i)29-s + (0.541 + 0.678i)37-s + (1.33 + 1.33i)41-s − 0.999i·45-s + (0.781 − 0.623i)49-s + (−0.900 + 1.43i)53-s + (1.59 − 0.559i)61-s + (1.00 − 1.59i)65-s + (1.90 + 0.433i)73-s + (−0.222 + 0.974i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2320\)    =    \(2^{4} \cdot 5 \cdot 29\)
Sign: $0.784 - 0.619i$
Analytic conductor: \(1.15783\)
Root analytic conductor: \(1.07602\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2320} (1743, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2320,\ (\ :0),\ 0.784 - 0.619i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.036875580\)
\(L(\frac12)\) \(\approx\) \(1.036875580\)
\(L(1)\) 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 \( 1 + (0.781 + 0.623i)T \)
29 \( 1 + (0.781 + 0.623i)T \)
good3 \( 1 + (-0.623 - 0.781i)T^{2} \)
7 \( 1 + (-0.781 + 0.623i)T^{2} \)
11 \( 1 + (0.974 - 0.222i)T^{2} \)
13 \( 1 + (-0.211 - 1.87i)T + (-0.974 + 0.222i)T^{2} \)
17 \( 1 + 0.445iT - T^{2} \)
19 \( 1 + (-0.781 - 0.623i)T^{2} \)
23 \( 1 + (-0.433 + 0.900i)T^{2} \)
31 \( 1 + (-0.433 - 0.900i)T^{2} \)
37 \( 1 + (-0.541 - 0.678i)T + (-0.222 + 0.974i)T^{2} \)
41 \( 1 + (-1.33 - 1.33i)T + iT^{2} \)
43 \( 1 + (0.900 + 0.433i)T^{2} \)
47 \( 1 + (0.222 + 0.974i)T^{2} \)
53 \( 1 + (0.900 - 1.43i)T + (-0.433 - 0.900i)T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (-1.59 + 0.559i)T + (0.781 - 0.623i)T^{2} \)
67 \( 1 + (0.974 + 0.222i)T^{2} \)
71 \( 1 + (-0.222 - 0.974i)T^{2} \)
73 \( 1 + (-1.90 - 0.433i)T + (0.900 + 0.433i)T^{2} \)
79 \( 1 + (0.974 + 0.222i)T^{2} \)
83 \( 1 + (-0.781 - 0.623i)T^{2} \)
89 \( 1 + (-1.05 + 1.68i)T + (-0.433 - 0.900i)T^{2} \)
97 \( 1 + (1.62 - 0.781i)T + (0.623 - 0.781i)T^{2} \)
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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

−9.330950453339572598535624550307, −8.446365777188569968629500470860, −7.72793778981727250505910773604, −7.09884278184248528900780446618, −6.24231501506850554212907137588, −5.08113324575349435818295695005, −4.41271792380512078789620648649, −3.85439692142547285669532824514, −2.41273856180707817060281610102, −1.33564074870340698000494528881, 0.808527199387539425724671641099, 2.47268035570756624707280087695, 3.54198594443047290609996824488, 3.93672083381781090002111122143, 5.21988430550303855016514660077, 6.02678347620048283737958884164, 6.88608234712598415581696774790, 7.60799848381889434301728536154, 8.157120858125783819939897285282, 9.094755600018557613281336230325

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