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Properties

Label 2-2300-5.4-c3-0-50
Degree $2$
Conductor $2300$
Sign $0.894 + 0.447i$
Analytic cond. $135.704$
Root an. cond. $11.6492$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.99i·3-s + 4.09i·7-s − 36.8·9-s − 47.4·11-s + 7.45i·13-s + 58.5i·17-s − 112.·19-s − 32.7·21-s − 23i·23-s − 78.7i·27-s − 51.2·29-s − 124.·31-s − 379. i·33-s + 366. i·37-s − 59.5·39-s + ⋯
L(s)  = 1  + 1.53i·3-s + 0.221i·7-s − 1.36·9-s − 1.30·11-s + 0.159i·13-s + 0.835i·17-s − 1.35·19-s − 0.340·21-s − 0.208i·23-s − 0.560i·27-s − 0.327·29-s − 0.719·31-s − 2.00i·33-s + 1.62i·37-s − 0.244·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2300\)    =    \(2^{2} \cdot 5^{2} \cdot 23\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(135.704\)
Root analytic conductor: \(11.6492\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{2300} (1749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2300,\ (\ :3/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1990229449\)
\(L(\frac12)\) \(\approx\) \(0.1990229449\)
\(L(\frac{5}{2})\) 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 \( 1 + 23iT \)
good3 \( 1 - 7.99iT - 27T^{2} \)
7 \( 1 - 4.09iT - 343T^{2} \)
11 \( 1 + 47.4T + 1.33e3T^{2} \)
13 \( 1 - 7.45iT - 2.19e3T^{2} \)
17 \( 1 - 58.5iT - 4.91e3T^{2} \)
19 \( 1 + 112.T + 6.85e3T^{2} \)
29 \( 1 + 51.2T + 2.43e4T^{2} \)
31 \( 1 + 124.T + 2.97e4T^{2} \)
37 \( 1 - 366. iT - 5.06e4T^{2} \)
41 \( 1 + 339.T + 6.89e4T^{2} \)
43 \( 1 - 497. iT - 7.95e4T^{2} \)
47 \( 1 + 609. iT - 1.03e5T^{2} \)
53 \( 1 + 120. iT - 1.48e5T^{2} \)
59 \( 1 - 309.T + 2.05e5T^{2} \)
61 \( 1 + 76.7T + 2.26e5T^{2} \)
67 \( 1 - 502. iT - 3.00e5T^{2} \)
71 \( 1 + 347.T + 3.57e5T^{2} \)
73 \( 1 - 99.0iT - 3.89e5T^{2} \)
79 \( 1 - 990.T + 4.93e5T^{2} \)
83 \( 1 + 71.3iT - 5.71e5T^{2} \)
89 \( 1 - 914.T + 7.04e5T^{2} \)
97 \( 1 + 258. iT - 9.12e5T^{2} \)
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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

−8.483079468560359637478073268272, −8.301433089081938713803547055299, −7.00496434995745601644085702188, −6.02451249885186331790189907385, −5.24815934553790573523017410956, −4.60766964643227938713298841619, −3.81472661900900225292273435729, −2.96153140780017524323070058303, −1.92834358944224217632583816818, −0.05234655176912022554434518348, 0.69413245686084860786877642209, 1.98951609297554081755101487073, 2.49939146007658353868274850469, 3.68845316558282498948031795044, 4.94638273813877181168775013491, 5.72296397387808304097336813913, 6.50998427016925796891516601163, 7.40239336923820369023003314567, 7.62379180293372217189478026781, 8.530295540514737488037690048635

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