Properties

Label 2-2925-5.4-c1-0-2
Degree $2$
Conductor $2925$
Sign $-0.894 + 0.447i$
Analytic cond. $23.3562$
Root an. cond. $4.83282$
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·2-s + 4-s − 2i·7-s + 3i·8-s − 4·11-s i·13-s + 2·14-s − 16-s + 4i·17-s − 6·19-s − 4i·22-s + 26-s − 2i·28-s + 4·29-s − 10·31-s + 5i·32-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.5·4-s − 0.755i·7-s + 1.06i·8-s − 1.20·11-s − 0.277i·13-s + 0.534·14-s − 0.250·16-s + 0.970i·17-s − 1.37·19-s − 0.852i·22-s + 0.196·26-s − 0.377i·28-s + 0.742·29-s − 1.79·31-s + 0.883i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2925\)    =    \(3^{2} \cdot 5^{2} \cdot 13\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(23.3562\)
Root analytic conductor: \(4.83282\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2925} (2224, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2925,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3900094338\)
\(L(\frac12)\) \(\approx\) \(0.3900094338\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 + iT \)
good2 \( 1 - iT - 2T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + 10T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 + 4iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 10iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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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

−8.842696837328822168070456740050, −8.260984650709569597645771600855, −7.60234051430765614882148630838, −7.00009446406438912127659312753, −6.20492204608251247810457418887, −5.53546138697624766168303037871, −4.68378257064995887687037458761, −3.69005198998887494817907210079, −2.62286568980532115017951900363, −1.68186261123444297073973559222, 0.10561412153367324341644205896, 1.75016680826516673529248393779, 2.50150024631790202981254708072, 3.16410410949470309381888198153, 4.29257128120266313789371610269, 5.22306153333215458357659172584, 6.00546090574766662132771827290, 6.85377203529618854808121645842, 7.53355356612732070911146823778, 8.432183105823413964062259850804

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