Properties

Label 2-693-1.1-c3-0-26
Degree $2$
Conductor $693$
Sign $1$
Analytic cond. $40.8883$
Root an. cond. $6.39439$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.44·2-s + 11.7·4-s + 22.0·5-s + 7·7-s − 16.6·8-s − 97.8·10-s − 11·11-s − 51.5·13-s − 31.1·14-s − 19.9·16-s + 26.5·17-s + 99.6·19-s + 258.·20-s + 48.8·22-s − 28.1·23-s + 359.·25-s + 229.·26-s + 82.2·28-s + 43.9·29-s − 83.8·31-s + 221.·32-s − 118.·34-s + 154.·35-s + 306.·37-s − 442.·38-s − 366.·40-s − 200.·41-s + ⋯
L(s)  = 1  − 1.57·2-s + 1.46·4-s + 1.96·5-s + 0.377·7-s − 0.736·8-s − 3.09·10-s − 0.301·11-s − 1.10·13-s − 0.593·14-s − 0.311·16-s + 0.378·17-s + 1.20·19-s + 2.89·20-s + 0.473·22-s − 0.255·23-s + 2.87·25-s + 1.72·26-s + 0.555·28-s + 0.281·29-s − 0.485·31-s + 1.22·32-s − 0.595·34-s + 0.744·35-s + 1.36·37-s − 1.89·38-s − 1.44·40-s − 0.765·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(40.8883\)
Root analytic conductor: \(6.39439\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.424556137\)
\(L(\frac12)\) \(\approx\) \(1.424556137\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 - 7T \)
11 \( 1 + 11T \)
good2 \( 1 + 4.44T + 8T^{2} \)
5 \( 1 - 22.0T + 125T^{2} \)
13 \( 1 + 51.5T + 2.19e3T^{2} \)
17 \( 1 - 26.5T + 4.91e3T^{2} \)
19 \( 1 - 99.6T + 6.85e3T^{2} \)
23 \( 1 + 28.1T + 1.21e4T^{2} \)
29 \( 1 - 43.9T + 2.43e4T^{2} \)
31 \( 1 + 83.8T + 2.97e4T^{2} \)
37 \( 1 - 306.T + 5.06e4T^{2} \)
41 \( 1 + 200.T + 6.89e4T^{2} \)
43 \( 1 + 13.7T + 7.95e4T^{2} \)
47 \( 1 - 266.T + 1.03e5T^{2} \)
53 \( 1 + 308.T + 1.48e5T^{2} \)
59 \( 1 - 622.T + 2.05e5T^{2} \)
61 \( 1 + 87.3T + 2.26e5T^{2} \)
67 \( 1 - 608.T + 3.00e5T^{2} \)
71 \( 1 - 464.T + 3.57e5T^{2} \)
73 \( 1 + 255.T + 3.89e5T^{2} \)
79 \( 1 - 261.T + 4.93e5T^{2} \)
83 \( 1 + 953.T + 5.71e5T^{2} \)
89 \( 1 - 839.T + 7.04e5T^{2} \)
97 \( 1 + 349.T + 9.12e5T^{2} \)
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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

−9.741841940155213333084476005904, −9.552887722902233487022643032276, −8.547530456429264241022975614183, −7.58793430162303772178531478675, −6.79230473259288939994217007188, −5.71884222139161686345200904551, −4.92934828404225892461492066174, −2.71850120161744464841518123254, −1.92126475152865460232851460853, −0.907942599976805648096740243873, 0.907942599976805648096740243873, 1.92126475152865460232851460853, 2.71850120161744464841518123254, 4.92934828404225892461492066174, 5.71884222139161686345200904551, 6.79230473259288939994217007188, 7.58793430162303772178531478675, 8.547530456429264241022975614183, 9.552887722902233487022643032276, 9.741841940155213333084476005904

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