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Properties

Label 2-46-23.22-c10-0-7
Degree $2$
Conductor $46$
Sign $0.178 + 0.983i$
Analytic cond. $29.2264$
Root an. cond. $5.40614$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 22.6·2-s − 356.·3-s + 512.·4-s − 2.60e3i·5-s + 8.05e3·6-s − 4.63e3i·7-s − 1.15e4·8-s + 6.77e4·9-s + 5.88e4i·10-s + 1.45e5i·11-s − 1.82e5·12-s − 1.62e5·13-s + 1.04e5i·14-s + 9.26e5i·15-s + 2.62e5·16-s + 1.84e6i·17-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.46·3-s + 0.500·4-s − 0.832i·5-s + 1.03·6-s − 0.275i·7-s − 0.353·8-s + 1.14·9-s + 0.588i·10-s + 0.901i·11-s − 0.732·12-s − 0.438·13-s + 0.195i·14-s + 1.22i·15-s + 0.250·16-s + 1.29i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.178 + 0.983i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.178 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(46\)    =    \(2 \cdot 23\)
Sign: $0.178 + 0.983i$
Analytic conductor: \(29.2264\)
Root analytic conductor: \(5.40614\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{46} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 46,\ (\ :5),\ 0.178 + 0.983i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.385684 - 0.321852i\)
\(L(\frac12)\) \(\approx\) \(0.385684 - 0.321852i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 22.6T \)
23 \( 1 + (1.15e6 + 6.33e6i)T \)
good3 \( 1 + 356.T + 5.90e4T^{2} \)
5 \( 1 + 2.60e3iT - 9.76e6T^{2} \)
7 \( 1 + 4.63e3iT - 2.82e8T^{2} \)
11 \( 1 - 1.45e5iT - 2.59e10T^{2} \)
13 \( 1 + 1.62e5T + 1.37e11T^{2} \)
17 \( 1 - 1.84e6iT - 2.01e12T^{2} \)
19 \( 1 - 2.46e6iT - 6.13e12T^{2} \)
29 \( 1 - 2.00e7T + 4.20e14T^{2} \)
31 \( 1 + 2.94e7T + 8.19e14T^{2} \)
37 \( 1 + 3.93e7iT - 4.80e15T^{2} \)
41 \( 1 + 2.96e7T + 1.34e16T^{2} \)
43 \( 1 + 6.64e6iT - 2.16e16T^{2} \)
47 \( 1 - 5.07e7T + 5.25e16T^{2} \)
53 \( 1 + 7.12e8iT - 1.74e17T^{2} \)
59 \( 1 + 7.18e8T + 5.11e17T^{2} \)
61 \( 1 - 1.73e8iT - 7.13e17T^{2} \)
67 \( 1 - 1.02e9iT - 1.82e18T^{2} \)
71 \( 1 + 9.59e7T + 3.25e18T^{2} \)
73 \( 1 - 3.16e8T + 4.29e18T^{2} \)
79 \( 1 - 1.66e9iT - 9.46e18T^{2} \)
83 \( 1 + 4.38e8iT - 1.55e19T^{2} \)
89 \( 1 + 5.78e9iT - 3.11e19T^{2} \)
97 \( 1 + 1.06e10iT - 7.37e19T^{2} \)
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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

−12.63379522012448495201807144826, −12.21314501298073174710431757873, −10.79001601593936722080710048862, −9.978218539804499644053122878036, −8.423237533136269629177830206584, −6.94244587453005906017836724494, −5.69339330527501416505228682048, −4.40898373180759062870726629597, −1.63759862517669763925855423225, −0.37001740501911080992138392663, 0.795199220880729977554585106215, 2.84700980988378302910190738865, 5.15243521828088599030760926423, 6.35895399708647433289312645133, 7.35393373887104941173238787238, 9.192027441029495476955866165144, 10.56195592455148990270969645022, 11.29085557890143253556525376518, 12.10887388702336107211495411253, 13.82531288472877865024771937710

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