Properties

Label 2-9702-1.1-c1-0-126
Degree $2$
Conductor $9702$
Sign $1$
Analytic cond. $77.4708$
Root an. cond. $8.80175$
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-s + 4-s + 3.29·5-s + 8-s + 3.29·10-s + 11-s + 6.06·13-s + 16-s + 6.11·17-s + 0.0511·19-s + 3.29·20-s + 22-s + 6.75·23-s + 5.82·25-s + 6.06·26-s − 2.82·29-s + 5.87·31-s + 32-s + 6.11·34-s − 8.31·37-s + 0.0511·38-s + 3.29·40-s − 6.11·41-s + 2.90·43-s + 44-s + 6.75·46-s − 1.22·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 1.47·5-s + 0.353·8-s + 1.04·10-s + 0.301·11-s + 1.68·13-s + 0.250·16-s + 1.48·17-s + 0.0117·19-s + 0.735·20-s + 0.213·22-s + 1.40·23-s + 1.16·25-s + 1.19·26-s − 0.525·29-s + 1.05·31-s + 0.176·32-s + 1.04·34-s − 1.36·37-s + 0.00830·38-s + 0.520·40-s − 0.955·41-s + 0.442·43-s + 0.150·44-s + 0.996·46-s − 0.178·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9702\)    =    \(2 \cdot 3^{2} \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(77.4708\)
Root analytic conductor: \(8.80175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9702,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.887328061\)
\(L(\frac12)\) \(\approx\) \(5.887328061\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 \)
11 \( 1 - T \)
good5 \( 1 - 3.29T + 5T^{2} \)
13 \( 1 - 6.06T + 13T^{2} \)
17 \( 1 - 6.11T + 17T^{2} \)
19 \( 1 - 0.0511T + 19T^{2} \)
23 \( 1 - 6.75T + 23T^{2} \)
29 \( 1 + 2.82T + 29T^{2} \)
31 \( 1 - 5.87T + 31T^{2} \)
37 \( 1 + 8.31T + 37T^{2} \)
41 \( 1 + 6.11T + 41T^{2} \)
43 \( 1 - 2.90T + 43T^{2} \)
47 \( 1 + 1.22T + 47T^{2} \)
53 \( 1 + 3.00T + 53T^{2} \)
59 \( 1 + 10.8T + 59T^{2} \)
61 \( 1 - 7.23T + 61T^{2} \)
67 \( 1 + 1.68T + 67T^{2} \)
71 \( 1 - 6.47T + 71T^{2} \)
73 \( 1 + 11.6T + 73T^{2} \)
79 \( 1 + 9.13T + 79T^{2} \)
83 \( 1 + 0.951T + 83T^{2} \)
89 \( 1 + 16.5T + 89T^{2} \)
97 \( 1 + 14.7T + 97T^{2} \)
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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

−7.50764303728098408264528315153, −6.67767397801707208629845845395, −6.26050793830290855650337918212, −5.51221735754222755811644187077, −5.24148545148005391033831629515, −4.17025833486356583580176413556, −3.33292651936476729224546006944, −2.80704125945089637808992115212, −1.56558271460816864309062141549, −1.25051474787283699224361739118, 1.25051474787283699224361739118, 1.56558271460816864309062141549, 2.80704125945089637808992115212, 3.33292651936476729224546006944, 4.17025833486356583580176413556, 5.24148545148005391033831629515, 5.51221735754222755811644187077, 6.26050793830290855650337918212, 6.67767397801707208629845845395, 7.50764303728098408264528315153

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