Properties

Label 2-2695-1.1-c1-0-62
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  − 2.31·2-s + 2.43·3-s + 3.33·4-s + 5-s − 5.62·6-s − 3.09·8-s + 2.92·9-s − 2.31·10-s − 11-s + 8.12·12-s + 3.20·13-s + 2.43·15-s + 0.468·16-s + 6.54·17-s − 6.75·18-s + 5.18·19-s + 3.33·20-s + 2.31·22-s − 0.628·23-s − 7.52·24-s + 25-s − 7.40·26-s − 0.188·27-s + 8.12·29-s − 5.62·30-s + 3.17·31-s + 5.10·32-s + ⋯
L(s)  = 1  − 1.63·2-s + 1.40·3-s + 1.66·4-s + 0.447·5-s − 2.29·6-s − 1.09·8-s + 0.974·9-s − 0.730·10-s − 0.301·11-s + 2.34·12-s + 0.889·13-s + 0.628·15-s + 0.117·16-s + 1.58·17-s − 1.59·18-s + 1.18·19-s + 0.746·20-s + 0.492·22-s − 0.131·23-s − 1.53·24-s + 0.200·25-s − 1.45·26-s − 0.0363·27-s + 1.50·29-s − 1.02·30-s + 0.569·31-s + 0.902·32-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

\(L(1)\) \(\approx\) \(1.696551842\)
\(L(\frac12)\) \(\approx\) \(1.696551842\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.31T + 2T^{2} \)
3 \( 1 - 2.43T + 3T^{2} \)
13 \( 1 - 3.20T + 13T^{2} \)
17 \( 1 - 6.54T + 17T^{2} \)
19 \( 1 - 5.18T + 19T^{2} \)
23 \( 1 + 0.628T + 23T^{2} \)
29 \( 1 - 8.12T + 29T^{2} \)
31 \( 1 - 3.17T + 31T^{2} \)
37 \( 1 + 11.7T + 37T^{2} \)
41 \( 1 + 4.12T + 41T^{2} \)
43 \( 1 + 4.61T + 43T^{2} \)
47 \( 1 - 2.44T + 47T^{2} \)
53 \( 1 + 7.23T + 53T^{2} \)
59 \( 1 - 6.05T + 59T^{2} \)
61 \( 1 + 0.349T + 61T^{2} \)
67 \( 1 - 2.18T + 67T^{2} \)
71 \( 1 + 3.53T + 71T^{2} \)
73 \( 1 + 1.72T + 73T^{2} \)
79 \( 1 + 14.5T + 79T^{2} \)
83 \( 1 - 14.7T + 83T^{2} \)
89 \( 1 + 5.45T + 89T^{2} \)
97 \( 1 - 19.4T + 97T^{2} \)
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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

−8.678104670508045728503766978531, −8.368359878688985637796941830902, −7.67805439173650309786917028201, −7.03978536907374074992604646453, −6.06580233977391592984575829073, −4.99667578744171461473264587142, −3.48010228315342660022175508496, −2.94379129351713899393564888335, −1.84406875703634420617952453683, −1.05267491498905046360166366469, 1.05267491498905046360166366469, 1.84406875703634420617952453683, 2.94379129351713899393564888335, 3.48010228315342660022175508496, 4.99667578744171461473264587142, 6.06580233977391592984575829073, 7.03978536907374074992604646453, 7.67805439173650309786917028201, 8.368359878688985637796941830902, 8.678104670508045728503766978531

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