Properties

Label 2-2695-1.1-c1-0-63
Degree $2$
Conductor $2695$
Sign $1$
Analytic cond. $21.5196$
Root an. cond. $4.63893$
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  + 1.21·2-s + 1.31·3-s − 0.525·4-s + 5-s + 1.59·6-s − 3.06·8-s − 1.28·9-s + 1.21·10-s + 11-s − 0.688·12-s + 2.68·13-s + 1.31·15-s − 2.67·16-s + 4.68·17-s − 1.55·18-s + 8.23·19-s − 0.525·20-s + 1.21·22-s − 4.14·23-s − 4.02·24-s + 25-s + 3.26·26-s − 5.61·27-s + 5.05·29-s + 1.59·30-s − 5.39·31-s + 2.88·32-s + ⋯
L(s)  = 1  + 0.858·2-s + 0.756·3-s − 0.262·4-s + 0.447·5-s + 0.649·6-s − 1.08·8-s − 0.426·9-s + 0.384·10-s + 0.301·11-s − 0.198·12-s + 0.745·13-s + 0.338·15-s − 0.668·16-s + 1.13·17-s − 0.366·18-s + 1.88·19-s − 0.117·20-s + 0.258·22-s − 0.864·23-s − 0.820·24-s + 0.200·25-s + 0.640·26-s − 1.08·27-s + 0.937·29-s + 0.290·30-s − 0.969·31-s + 0.510·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2695\)    =    \(5 \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(21.5196\)
Root analytic conductor: \(4.63893\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2695,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.545148767\)
\(L(\frac12)\) \(\approx\) \(3.545148767\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
7 \( 1 \)
11 \( 1 - T \)
good2 \( 1 - 1.21T + 2T^{2} \)
3 \( 1 - 1.31T + 3T^{2} \)
13 \( 1 - 2.68T + 13T^{2} \)
17 \( 1 - 4.68T + 17T^{2} \)
19 \( 1 - 8.23T + 19T^{2} \)
23 \( 1 + 4.14T + 23T^{2} \)
29 \( 1 - 5.05T + 29T^{2} \)
31 \( 1 + 5.39T + 31T^{2} \)
37 \( 1 - 4.76T + 37T^{2} \)
41 \( 1 - 1.16T + 41T^{2} \)
43 \( 1 - 5.95T + 43T^{2} \)
47 \( 1 - 8.68T + 47T^{2} \)
53 \( 1 + 0.769T + 53T^{2} \)
59 \( 1 + 4.02T + 59T^{2} \)
61 \( 1 - 0.407T + 61T^{2} \)
67 \( 1 + 13.1T + 67T^{2} \)
71 \( 1 + 2.19T + 71T^{2} \)
73 \( 1 - 10.9T + 73T^{2} \)
79 \( 1 - 8.51T + 79T^{2} \)
83 \( 1 - 9.90T + 83T^{2} \)
89 \( 1 - 9.80T + 89T^{2} \)
97 \( 1 + 8.85T + 97T^{2} \)
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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

−9.022860087613765962016069408275, −8.067360394815936032060835518047, −7.46540798749994557457233480184, −6.13698867291239538994382008555, −5.75292630469813870874575019884, −4.96490222140628190201365359973, −3.83314358220441212368194207507, −3.33110215329385619342542164004, −2.50075049751627098615027948491, −1.05955041249915747520289383113, 1.05955041249915747520289383113, 2.50075049751627098615027948491, 3.33110215329385619342542164004, 3.83314358220441212368194207507, 4.96490222140628190201365359973, 5.75292630469813870874575019884, 6.13698867291239538994382008555, 7.46540798749994557457233480184, 8.067360394815936032060835518047, 9.022860087613765962016069408275

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