Properties

Label 2-2900-1.1-c1-0-17
Degree $2$
Conductor $2900$
Sign $1$
Analytic cond. $23.1566$
Root an. cond. $4.81213$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 4·7-s + 6·9-s − 11-s + 3·13-s − 2·17-s + 4·19-s − 12·21-s + 6·23-s + 9·27-s − 29-s + 9·31-s − 3·33-s + 8·37-s + 9·39-s − 8·41-s + 5·43-s + 7·47-s + 9·49-s − 6·51-s + 5·53-s + 12·57-s − 10·59-s + 10·61-s − 24·63-s − 8·67-s + 18·69-s + ⋯
L(s)  = 1  + 1.73·3-s − 1.51·7-s + 2·9-s − 0.301·11-s + 0.832·13-s − 0.485·17-s + 0.917·19-s − 2.61·21-s + 1.25·23-s + 1.73·27-s − 0.185·29-s + 1.61·31-s − 0.522·33-s + 1.31·37-s + 1.44·39-s − 1.24·41-s + 0.762·43-s + 1.02·47-s + 9/7·49-s − 0.840·51-s + 0.686·53-s + 1.58·57-s − 1.30·59-s + 1.28·61-s − 3.02·63-s − 0.977·67-s + 2.16·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2900\)    =    \(2^{2} \cdot 5^{2} \cdot 29\)
Sign: $1$
Analytic conductor: \(23.1566\)
Root analytic conductor: \(4.81213\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2900,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.201218412\)
\(L(\frac12)\) \(\approx\) \(3.201218412\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
5 \( 1 \)
29 \( 1 + T \)
good3 \( 1 - p T + p T^{2} \) 1.3.ad
7 \( 1 + 4 T + p T^{2} \) 1.7.e
11 \( 1 + T + p T^{2} \) 1.11.b
13 \( 1 - 3 T + p T^{2} \) 1.13.ad
17 \( 1 + 2 T + p T^{2} \) 1.17.c
19 \( 1 - 4 T + p T^{2} \) 1.19.ae
23 \( 1 - 6 T + p T^{2} \) 1.23.ag
31 \( 1 - 9 T + p T^{2} \) 1.31.aj
37 \( 1 - 8 T + p T^{2} \) 1.37.ai
41 \( 1 + 8 T + p T^{2} \) 1.41.i
43 \( 1 - 5 T + p T^{2} \) 1.43.af
47 \( 1 - 7 T + p T^{2} \) 1.47.ah
53 \( 1 - 5 T + p T^{2} \) 1.53.af
59 \( 1 + 10 T + p T^{2} \) 1.59.k
61 \( 1 - 10 T + p T^{2} \) 1.61.ak
67 \( 1 + 8 T + p T^{2} \) 1.67.i
71 \( 1 + 2 T + p T^{2} \) 1.71.c
73 \( 1 + p T^{2} \) 1.73.a
79 \( 1 + T + p T^{2} \) 1.79.b
83 \( 1 + 6 T + p T^{2} \) 1.83.g
89 \( 1 - 12 T + p T^{2} \) 1.89.am
97 \( 1 + p T^{2} \) 1.97.a
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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

−8.879395748073981170118839519763, −8.141025642923753279087882974777, −7.33550264153574239993784581896, −6.71069646532965173957055154462, −5.86667719704416489044012817015, −4.57978403736801959392711577274, −3.68228666659704414023251144515, −3.04081723023151694497231831115, −2.50541091856436250305362274671, −1.05600136943034290088228750938, 1.05600136943034290088228750938, 2.50541091856436250305362274671, 3.04081723023151694497231831115, 3.68228666659704414023251144515, 4.57978403736801959392711577274, 5.86667719704416489044012817015, 6.71069646532965173957055154462, 7.33550264153574239993784581896, 8.141025642923753279087882974777, 8.879395748073981170118839519763

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