Properties

Label 2-46-1.1-c3-0-0
Degree $2$
Conductor $46$
Sign $1$
Analytic cond. $2.71408$
Root an. cond. $1.64744$
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  − 2·2-s − 10.1·3-s + 4·4-s + 11.4·5-s + 20.2·6-s + 9.40·7-s − 8·8-s + 75.1·9-s − 22.8·10-s + 10.3·11-s − 40.4·12-s − 33.0·13-s − 18.8·14-s − 115.·15-s + 16·16-s + 138.·17-s − 150.·18-s + 68.6·19-s + 45.6·20-s − 95.0·21-s − 20.7·22-s − 23·23-s + 80.8·24-s + 5.03·25-s + 66.1·26-s − 486.·27-s + 37.6·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.94·3-s + 0.5·4-s + 1.01·5-s + 1.37·6-s + 0.507·7-s − 0.353·8-s + 2.78·9-s − 0.721·10-s + 0.284·11-s − 0.972·12-s − 0.705·13-s − 0.359·14-s − 1.98·15-s + 0.250·16-s + 1.98·17-s − 1.96·18-s + 0.828·19-s + 0.509·20-s − 0.987·21-s − 0.201·22-s − 0.208·23-s + 0.687·24-s + 0.0402·25-s + 0.499·26-s − 3.46·27-s + 0.253·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(46\)    =    \(2 \cdot 23\)
Sign: $1$
Analytic conductor: \(2.71408\)
Root analytic conductor: \(1.64744\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 46,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6944059846\)
\(L(\frac12)\) \(\approx\) \(0.6944059846\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2T \)
23 \( 1 + 23T \)
good3 \( 1 + 10.1T + 27T^{2} \)
5 \( 1 - 11.4T + 125T^{2} \)
7 \( 1 - 9.40T + 343T^{2} \)
11 \( 1 - 10.3T + 1.33e3T^{2} \)
13 \( 1 + 33.0T + 2.19e3T^{2} \)
17 \( 1 - 138.T + 4.91e3T^{2} \)
19 \( 1 - 68.6T + 6.85e3T^{2} \)
29 \( 1 - 215.T + 2.43e4T^{2} \)
31 \( 1 - 87.9T + 2.97e4T^{2} \)
37 \( 1 + 215.T + 5.06e4T^{2} \)
41 \( 1 - 175.T + 6.89e4T^{2} \)
43 \( 1 + 40.7T + 7.95e4T^{2} \)
47 \( 1 - 405.T + 1.03e5T^{2} \)
53 \( 1 + 276.T + 1.48e5T^{2} \)
59 \( 1 - 293.T + 2.05e5T^{2} \)
61 \( 1 + 450.T + 2.26e5T^{2} \)
67 \( 1 - 273.T + 3.00e5T^{2} \)
71 \( 1 + 643.T + 3.57e5T^{2} \)
73 \( 1 - 106.T + 3.89e5T^{2} \)
79 \( 1 - 60.0T + 4.93e5T^{2} \)
83 \( 1 + 372.T + 5.71e5T^{2} \)
89 \( 1 + 543.T + 7.04e5T^{2} \)
97 \( 1 - 550.T + 9.12e5T^{2} \)
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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

−15.83766390324144604742283554838, −14.14581142762574622075170632334, −12.39128154989715565409589521915, −11.70641608737296900657507054495, −10.34956937814400208999460499009, −9.770220985783675735041913252013, −7.47958367205089572145023962826, −6.11514973417687853277953235215, −5.13763445710730388341946220202, −1.21849296306196614015931430381, 1.21849296306196614015931430381, 5.13763445710730388341946220202, 6.11514973417687853277953235215, 7.47958367205089572145023962826, 9.770220985783675735041913252013, 10.34956937814400208999460499009, 11.70641608737296900657507054495, 12.39128154989715565409589521915, 14.14581142762574622075170632334, 15.83766390324144604742283554838

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