Properties

Label 2-2400-40.29-c1-0-10
Degree $2$
Conductor $2400$
Sign $0.912 - 0.409i$
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  − 3-s − 1.33i·7-s + 9-s − 2.94i·11-s − 2.04·13-s + 3.61i·17-s + 5.35i·19-s + 1.33i·21-s + 8.59i·23-s − 27-s − 5.26i·29-s + 2.08·31-s + 2.94i·33-s − 6.55·37-s + 2.04·39-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.504i·7-s + 0.333·9-s − 0.887i·11-s − 0.566·13-s + 0.876i·17-s + 1.22i·19-s + 0.291i·21-s + 1.79i·23-s − 0.192·27-s − 0.977i·29-s + 0.373·31-s + 0.512i·33-s − 1.07·37-s + 0.326·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 - 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
Sign: $0.912 - 0.409i$
Analytic conductor: \(19.1640\)
Root analytic conductor: \(4.37768\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2400} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2400,\ (\ :1/2),\ 0.912 - 0.409i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.257423235\)
\(L(\frac12)\) \(\approx\) \(1.257423235\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 \)
good7 \( 1 + 1.33iT - 7T^{2} \)
11 \( 1 + 2.94iT - 11T^{2} \)
13 \( 1 + 2.04T + 13T^{2} \)
17 \( 1 - 3.61iT - 17T^{2} \)
19 \( 1 - 5.35iT - 19T^{2} \)
23 \( 1 - 8.59iT - 23T^{2} \)
29 \( 1 + 5.26iT - 29T^{2} \)
31 \( 1 - 2.08T + 31T^{2} \)
37 \( 1 + 6.55T + 37T^{2} \)
41 \( 1 - 7.02T + 41T^{2} \)
43 \( 1 - 8.50T + 43T^{2} \)
47 \( 1 + 9.97iT - 47T^{2} \)
53 \( 1 + 6.12T + 53T^{2} \)
59 \( 1 - 4.75iT - 59T^{2} \)
61 \( 1 + 8.51iT - 61T^{2} \)
67 \( 1 - 10.6T + 67T^{2} \)
71 \( 1 - 2.62T + 71T^{2} \)
73 \( 1 - 15.3iT - 73T^{2} \)
79 \( 1 - 10.4T + 79T^{2} \)
83 \( 1 + 1.52T + 83T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + 13.4iT - 97T^{2} \)
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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

−9.078546233040213713801571507980, −8.043645131557329235431292735998, −7.58777881751358653527267217572, −6.60066555626826331180497649387, −5.82530688320974186596082834773, −5.27913438503734149175477815196, −4.04549777125763973124246757794, −3.53415583092980473157665649614, −2.07685977307812227186517867842, −0.884158086113691994082701455096, 0.61616521460414197998945198697, 2.17345160636381986864073793722, 2.92615821375751669960269595894, 4.47647350117957864787646549868, 4.81229805524614347119843088858, 5.74073714086087992853263206125, 6.73319422291286091196335157319, 7.15301513214336900643467674472, 8.100835345590996190316141029390, 9.177224032386892197289962750086

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