Properties

Label 2-2400-8.5-c1-0-2
Degree $2$
Conductor $2400$
Sign $0.102 - 0.994i$
Analytic cond. $19.1640$
Root an. cond. $4.37768$
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  i·3-s − 3.62·7-s − 9-s − 6.20i·11-s + 0.578i·13-s − 1.42·17-s + 5.62i·19-s + 3.62i·21-s − 5.62·23-s + i·27-s + 2i·29-s + 2.57·31-s − 6.20·33-s + 7.83i·37-s + 0.578·39-s + ⋯
L(s)  = 1  − 0.577i·3-s − 1.37·7-s − 0.333·9-s − 1.87i·11-s + 0.160i·13-s − 0.344·17-s + 1.29i·19-s + 0.791i·21-s − 1.17·23-s + 0.192i·27-s + 0.371i·29-s + 0.463·31-s − 1.08·33-s + 1.28i·37-s + 0.0926·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.102 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.102 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
Sign: $0.102 - 0.994i$
Analytic conductor: \(19.1640\)
Root analytic conductor: \(4.37768\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2400} (1201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2400,\ (\ :1/2),\ 0.102 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4867042716\)
\(L(\frac12)\) \(\approx\) \(0.4867042716\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + iT \)
5 \( 1 \)
good7 \( 1 + 3.62T + 7T^{2} \)
11 \( 1 + 6.20iT - 11T^{2} \)
13 \( 1 - 0.578iT - 13T^{2} \)
17 \( 1 + 1.42T + 17T^{2} \)
19 \( 1 - 5.62iT - 19T^{2} \)
23 \( 1 + 5.62T + 23T^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 - 2.57T + 31T^{2} \)
37 \( 1 - 7.83iT - 37T^{2} \)
41 \( 1 - 5.25T + 41T^{2} \)
43 \( 1 + 7.25iT - 43T^{2} \)
47 \( 1 - 6.78T + 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 - 2.20iT - 59T^{2} \)
61 \( 1 - 12.4iT - 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 8.41T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + 5.42T + 79T^{2} \)
83 \( 1 - 3.25iT - 83T^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 + 4.84T + 97T^{2} \)
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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

−8.936175423526271544547350594860, −8.460642773472302202619961707654, −7.61932783802440706564402959468, −6.68239212768573033227118853677, −6.01414255265054616768199200942, −5.66918936784977649085150449847, −4.07344265595133827019981963258, −3.35548540186479242585236426842, −2.53740524689382711968890640842, −1.07728734759227355995901152384, 0.18035925907624018540457071447, 2.13118871086503861692011518589, 2.94652229054856263313718280826, 4.08559552049577346898908031652, 4.58987481826807309364550118405, 5.66788001808844737142903384388, 6.52547095115998231957659661253, 7.13595952867525701155580329340, 7.978742550302285725701720618465, 9.145886679968846785439125403841

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