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Properties

Label 2-2320-145.28-c0-0-0
Degree $2$
Conductor $2320$
Sign $0.850 + 0.525i$
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  − i·5-s + (1 − i)7-s + i·9-s + (1 + i)13-s + (1 + i)23-s − 25-s − i·29-s + (−1 − i)35-s + 45-s − i·49-s + (−1 − i)53-s − 2i·59-s + (1 + i)63-s + (1 − i)65-s + (−1 + i)67-s + ⋯
L(s)  = 1  − i·5-s + (1 − i)7-s + i·9-s + (1 + i)13-s + (1 + i)23-s − 25-s − i·29-s + (−1 − i)35-s + 45-s − i·49-s + (−1 − i)53-s − 2i·59-s + (1 + i)63-s + (1 − i)65-s + (−1 + i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2320\)    =    \(2^{4} \cdot 5 \cdot 29\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(1.15783\)
Root analytic conductor: \(1.07602\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2320} (753, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2320,\ (\ :0),\ 0.850 + 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.401811179\)
\(L(\frac12)\) \(\approx\) \(1.401811179\)
\(L(1)\) 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 \( 1 + iT \)
29 \( 1 + iT \)
good3 \( 1 - iT^{2} \)
7 \( 1 + (-1 + i)T - iT^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (-1 - i)T + iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-1 - i)T + iT^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (1 + i)T + iT^{2} \)
59 \( 1 + 2iT - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (1 - i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (1 + i)T + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.020120120962431484528078601030, −8.247372046037248566864009681600, −7.76710278560610859414036569307, −6.95260477128519008202926491481, −5.85621027300197611401860105262, −4.93573268044737551267005094315, −4.47420202041200721169318549589, −3.61283062905507850958908177967, −1.95213025970177672999325147755, −1.23695605757599285905606199819, 1.36727079939141460685565965729, 2.70974029638537184549547641060, 3.29673707933963177596307621610, 4.43042095618400575822405445764, 5.51086502258874555286809513481, 6.08971300989456679638358876554, 6.88164169238337573858967322056, 7.74737665535135326952538920317, 8.612465828442057916484860874500, 9.015432583997362538446415521958

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