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Properties

Label 2-2300-5.4-c3-0-22
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  + 3.31i·3-s + 35.2i·7-s + 16.0·9-s + 35.8·11-s + 26.4i·13-s + 92.3i·17-s − 71.9·19-s − 116.·21-s − 23i·23-s + 142. i·27-s − 131.·29-s − 330.·31-s + 118. i·33-s + 55.6i·37-s − 87.7·39-s + ⋯
L(s)  = 1  + 0.637i·3-s + 1.90i·7-s + 0.593·9-s + 0.981·11-s + 0.564i·13-s + 1.31i·17-s − 0.868·19-s − 1.21·21-s − 0.208i·23-s + 1.01i·27-s − 0.843·29-s − 1.91·31-s + 0.626i·33-s + 0.247i·37-s − 0.360·39-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

\(L(2)\) \(\approx\) \(1.644316685\)
\(L(\frac12)\) \(\approx\) \(1.644316685\)
\(L(\frac{5}{2})\) 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 \( 1 - 3.31iT - 27T^{2} \)
7 \( 1 - 35.2iT - 343T^{2} \)
11 \( 1 - 35.8T + 1.33e3T^{2} \)
13 \( 1 - 26.4iT - 2.19e3T^{2} \)
17 \( 1 - 92.3iT - 4.91e3T^{2} \)
19 \( 1 + 71.9T + 6.85e3T^{2} \)
29 \( 1 + 131.T + 2.43e4T^{2} \)
31 \( 1 + 330.T + 2.97e4T^{2} \)
37 \( 1 - 55.6iT - 5.06e4T^{2} \)
41 \( 1 - 403.T + 6.89e4T^{2} \)
43 \( 1 + 289. iT - 7.95e4T^{2} \)
47 \( 1 - 141. iT - 1.03e5T^{2} \)
53 \( 1 - 0.542iT - 1.48e5T^{2} \)
59 \( 1 - 241.T + 2.05e5T^{2} \)
61 \( 1 + 779.T + 2.26e5T^{2} \)
67 \( 1 - 324. iT - 3.00e5T^{2} \)
71 \( 1 - 408.T + 3.57e5T^{2} \)
73 \( 1 - 124. iT - 3.89e5T^{2} \)
79 \( 1 - 896.T + 4.93e5T^{2} \)
83 \( 1 - 508. iT - 5.71e5T^{2} \)
89 \( 1 - 908.T + 7.04e5T^{2} \)
97 \( 1 - 549. iT - 9.12e5T^{2} \)
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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

−9.102242879147742182958551771279, −8.684377138621383306168984652144, −7.63283826971773222728082612927, −6.55095307437814245493011583309, −5.96613389027112338871856644139, −5.18936543902609130962871997581, −4.17916394644234212287626811082, −3.59917064271969756366155378660, −2.23208199099577972880030134562, −1.62767622581435543115874254314, 0.34004748075073622184532496676, 1.05752835792279089782975931092, 1.98021358515102109284336244039, 3.44148085115144353819739422315, 4.08733000266209100179720302832, 4.85669188960406693346365259934, 6.12541066277081842302498174551, 6.85410222287255245805829278788, 7.48723723668728939967973929845, 7.76113813157495858016709821238

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