Properties

Label 2-5043-1.1-c1-0-11
Degree $2$
Conductor $5043$
Sign $1$
Analytic cond. $40.2685$
Root an. cond. $6.34575$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.81·2-s + 3-s + 1.28·4-s − 1.10·5-s − 1.81·6-s − 2.52·7-s + 1.28·8-s + 9-s + 2·10-s − 0.813·11-s + 1.28·12-s − 5.10·13-s + 4.57·14-s − 1.10·15-s − 4.91·16-s − 3.39·17-s − 1.81·18-s − 3.10·19-s − 1.42·20-s − 2.52·21-s + 1.47·22-s − 0.897·23-s + 1.28·24-s − 3.78·25-s + 9.25·26-s + 27-s − 3.25·28-s + ⋯
L(s)  = 1  − 1.28·2-s + 0.577·3-s + 0.644·4-s − 0.493·5-s − 0.740·6-s − 0.954·7-s + 0.455·8-s + 0.333·9-s + 0.632·10-s − 0.245·11-s + 0.372·12-s − 1.41·13-s + 1.22·14-s − 0.284·15-s − 1.22·16-s − 0.822·17-s − 0.427·18-s − 0.711·19-s − 0.317·20-s − 0.550·21-s + 0.314·22-s − 0.187·23-s + 0.263·24-s − 0.756·25-s + 1.81·26-s + 0.192·27-s − 0.615·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5043\)    =    \(3 \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(40.2685\)
Root analytic conductor: \(6.34575\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5043,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2485354381\)
\(L(\frac12)\) \(\approx\) \(0.2485354381\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
41 \( 1 \)
good2 \( 1 + 1.81T + 2T^{2} \)
5 \( 1 + 1.10T + 5T^{2} \)
7 \( 1 + 2.52T + 7T^{2} \)
11 \( 1 + 0.813T + 11T^{2} \)
13 \( 1 + 5.10T + 13T^{2} \)
17 \( 1 + 3.39T + 17T^{2} \)
19 \( 1 + 3.10T + 19T^{2} \)
23 \( 1 + 0.897T + 23T^{2} \)
29 \( 1 + 4.44T + 29T^{2} \)
31 \( 1 + 8.96T + 31T^{2} \)
37 \( 1 - 2.08T + 37T^{2} \)
43 \( 1 - 9.07T + 43T^{2} \)
47 \( 1 - 0.235T + 47T^{2} \)
53 \( 1 + 13.8T + 53T^{2} \)
59 \( 1 - 1.04T + 59T^{2} \)
61 \( 1 - 1.91T + 61T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 - 12.8T + 71T^{2} \)
73 \( 1 - 4.75T + 73T^{2} \)
79 \( 1 - 15.8T + 79T^{2} \)
83 \( 1 + 7.68T + 83T^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 + 2.72T + 97T^{2} \)
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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

−8.168000589302083366251879817517, −7.74357613824643311468506296832, −7.09604566880089604022175709846, −6.49035664974094224596680695045, −5.31849849066486256569340005611, −4.34990327866311890218572410931, −3.68086830236714436941611763717, −2.52859475688185224850416808999, −1.91883547711025957690754122727, −0.30180690602701642549227551081, 0.30180690602701642549227551081, 1.91883547711025957690754122727, 2.52859475688185224850416808999, 3.68086830236714436941611763717, 4.34990327866311890218572410931, 5.31849849066486256569340005611, 6.49035664974094224596680695045, 7.09604566880089604022175709846, 7.74357613824643311468506296832, 8.168000589302083366251879817517

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