Properties

Label 2-2300-5.4-c1-0-29
Degree $2$
Conductor $2300$
Sign $-0.894 - 0.447i$
Analytic cond. $18.3655$
Root an. cond. $4.28550$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.56i·3-s − 1.56i·7-s − 3.56·9-s + 2·11-s − 0.561i·13-s − 1.56i·17-s − 6·19-s − 4·21-s i·23-s + 1.43i·27-s + 2.12·29-s − 9.24·31-s − 5.12i·33-s − 0.438i·37-s − 1.43·39-s + ⋯
L(s)  = 1  − 1.47i·3-s − 0.590i·7-s − 1.18·9-s + 0.603·11-s − 0.155i·13-s − 0.378i·17-s − 1.37·19-s − 0.872·21-s − 0.208i·23-s + 0.276i·27-s + 0.394·29-s − 1.66·31-s − 0.891i·33-s − 0.0720i·37-s − 0.230·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2300\)    =    \(2^{2} \cdot 5^{2} \cdot 23\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(18.3655\)
Root analytic conductor: \(4.28550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2300} (1749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2300,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9882900182\)
\(L(\frac12)\) \(\approx\) \(0.9882900182\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23 \( 1 + iT \)
good3 \( 1 + 2.56iT - 3T^{2} \)
7 \( 1 + 1.56iT - 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 0.561iT - 13T^{2} \)
17 \( 1 + 1.56iT - 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
29 \( 1 - 2.12T + 29T^{2} \)
31 \( 1 + 9.24T + 31T^{2} \)
37 \( 1 + 0.438iT - 37T^{2} \)
41 \( 1 + 4.12T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 7.68iT - 47T^{2} \)
53 \( 1 - 0.438iT - 53T^{2} \)
59 \( 1 + 8.68T + 59T^{2} \)
61 \( 1 - 1.12T + 61T^{2} \)
67 \( 1 + 4.43iT - 67T^{2} \)
71 \( 1 - 1.87T + 71T^{2} \)
73 \( 1 - 8.56iT - 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 + 14.9iT - 83T^{2} \)
89 \( 1 - 2.24T + 89T^{2} \)
97 \( 1 + 4.87iT - 97T^{2} \)
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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

−8.465520580123938441425438138384, −7.61928346212063383704790655859, −7.02085720165371815436096533349, −6.47720705641153756255839716199, −5.66242896332010733462295868651, −4.50418075742600502692610010654, −3.54952314532672067303804848579, −2.32894006910848013716554516161, −1.49522588001476367536538684544, −0.32703884585831648308667288333, 1.81272631460850310706014414490, 3.02932428616861243646637013049, 3.94674267463688639654017916825, 4.49639262794295976712704068816, 5.43466003893453319520443709987, 6.12349114470830002877418782494, 7.05959873346787301227638776622, 8.236468469072239596160313732725, 8.911775451305974453188708903889, 9.362189695739923734763079104641

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