Properties

Label 2-5290-1.1-c1-0-141
Degree $2$
Conductor $5290$
Sign $-1$
Analytic cond. $42.2408$
Root an. cond. $6.49929$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 0.551·3-s + 4-s − 5-s − 0.551·6-s + 3.69·7-s + 8-s − 2.69·9-s − 10-s − 1.44·11-s − 0.551·12-s − 5.69·13-s + 3.69·14-s + 0.551·15-s + 16-s − 4.55·17-s − 2.69·18-s + 6.79·19-s − 20-s − 2.03·21-s − 1.44·22-s − 0.551·24-s + 25-s − 5.69·26-s + 3.14·27-s + 3.69·28-s + 10.4·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.318·3-s + 0.5·4-s − 0.447·5-s − 0.225·6-s + 1.39·7-s + 0.353·8-s − 0.898·9-s − 0.316·10-s − 0.436·11-s − 0.159·12-s − 1.57·13-s + 0.987·14-s + 0.142·15-s + 0.250·16-s − 1.10·17-s − 0.635·18-s + 1.55·19-s − 0.223·20-s − 0.445·21-s − 0.308·22-s − 0.112·24-s + 0.200·25-s − 1.11·26-s + 0.604·27-s + 0.698·28-s + 1.94·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5290\)    =    \(2 \cdot 5 \cdot 23^{2}\)
Sign: $-1$
Analytic conductor: \(42.2408\)
Root analytic conductor: \(6.49929\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5290} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5290,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 + T \)
23 \( 1 \)
good3 \( 1 + 0.551T + 3T^{2} \)
7 \( 1 - 3.69T + 7T^{2} \)
11 \( 1 + 1.44T + 11T^{2} \)
13 \( 1 + 5.69T + 13T^{2} \)
17 \( 1 + 4.55T + 17T^{2} \)
19 \( 1 - 6.79T + 19T^{2} \)
29 \( 1 - 10.4T + 29T^{2} \)
31 \( 1 + 2.59T + 31T^{2} \)
37 \( 1 + 0.207T + 37T^{2} \)
41 \( 1 - 1.44T + 41T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 + 13.3T + 53T^{2} \)
59 \( 1 - 4.49T + 59T^{2} \)
61 \( 1 + 11.7T + 61T^{2} \)
67 \( 1 - 3.10T + 67T^{2} \)
71 \( 1 + 4.55T + 71T^{2} \)
73 \( 1 - 2.49T + 73T^{2} \)
79 \( 1 + 10.2T + 79T^{2} \)
83 \( 1 + 7.59T + 83T^{2} \)
89 \( 1 - 7.39T + 89T^{2} \)
97 \( 1 - 10.3T + 97T^{2} \)
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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

−7.895337797287901256158958447114, −7.07813616535049958922285038028, −6.37808478109772162192608334779, −5.25787114561030642735006405671, −4.96789176554309532420247198927, −4.49000056674169312895275447553, −3.16795434838478300119454357167, −2.58464313245200089056499169978, −1.49995593021040348811320882232, 0, 1.49995593021040348811320882232, 2.58464313245200089056499169978, 3.16795434838478300119454357167, 4.49000056674169312895275447553, 4.96789176554309532420247198927, 5.25787114561030642735006405671, 6.37808478109772162192608334779, 7.07813616535049958922285038028, 7.895337797287901256158958447114

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