Properties

Label 2-46-1.1-c3-0-1
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 + 9.10·3-s + 4·4-s − 1.40·5-s − 18.2·6-s − 3.40·7-s − 8·8-s + 55.8·9-s + 2.80·10-s + 61.6·11-s + 36.4·12-s − 77.9·13-s + 6.80·14-s − 12.7·15-s + 16·16-s − 14.8·17-s − 111.·18-s − 46.6·19-s − 5.61·20-s − 30.9·21-s − 123.·22-s − 23·23-s − 72.8·24-s − 123.·25-s + 155.·26-s + 263.·27-s − 13.6·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.75·3-s + 0.5·4-s − 0.125·5-s − 1.23·6-s − 0.183·7-s − 0.353·8-s + 2.07·9-s + 0.0887·10-s + 1.68·11-s + 0.876·12-s − 1.66·13-s + 0.129·14-s − 0.219·15-s + 0.250·16-s − 0.211·17-s − 1.46·18-s − 0.563·19-s − 0.0627·20-s − 0.321·21-s − 1.19·22-s − 0.208·23-s − 0.619·24-s − 0.984·25-s + 1.17·26-s + 1.87·27-s − 0.0918·28-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

\(L(2)\) \(\approx\) \(1.579889549\)
\(L(\frac12)\) \(\approx\) \(1.579889549\)
\(L(\frac{5}{2})\) 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 \( 1 - 9.10T + 27T^{2} \)
5 \( 1 + 1.40T + 125T^{2} \)
7 \( 1 + 3.40T + 343T^{2} \)
11 \( 1 - 61.6T + 1.33e3T^{2} \)
13 \( 1 + 77.9T + 2.19e3T^{2} \)
17 \( 1 + 14.8T + 4.91e3T^{2} \)
19 \( 1 + 46.6T + 6.85e3T^{2} \)
29 \( 1 + 200.T + 2.43e4T^{2} \)
31 \( 1 + 20.9T + 2.97e4T^{2} \)
37 \( 1 - 233.T + 5.06e4T^{2} \)
41 \( 1 - 309.T + 6.89e4T^{2} \)
43 \( 1 + 399.T + 7.95e4T^{2} \)
47 \( 1 + 190.T + 1.03e5T^{2} \)
53 \( 1 - 516.T + 1.48e5T^{2} \)
59 \( 1 - 498.T + 2.05e5T^{2} \)
61 \( 1 - 906.T + 2.26e5T^{2} \)
67 \( 1 + 33.6T + 3.00e5T^{2} \)
71 \( 1 + 61.1T + 3.57e5T^{2} \)
73 \( 1 + 79.3T + 3.89e5T^{2} \)
79 \( 1 - 533.T + 4.93e5T^{2} \)
83 \( 1 - 766.T + 5.71e5T^{2} \)
89 \( 1 - 57.9T + 7.04e5T^{2} \)
97 \( 1 - 1.60e3T + 9.12e5T^{2} \)
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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

−14.85263696454764746462251854751, −14.62068915426045721494616393094, −13.11628615936547593268272151830, −11.76206334562147309088571858946, −9.805456643526772455206132982894, −9.236446159925764464203543002579, −8.008348909776890027069974770128, −6.90305430364277925270365166073, −3.89321767094004652155698373724, −2.15045492077248343385994039764, 2.15045492077248343385994039764, 3.89321767094004652155698373724, 6.90305430364277925270365166073, 8.008348909776890027069974770128, 9.236446159925764464203543002579, 9.805456643526772455206132982894, 11.76206334562147309088571858946, 13.11628615936547593268272151830, 14.62068915426045721494616393094, 14.85263696454764746462251854751

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