Properties

Label 2-3-1.1-c43-0-3
Degree $2$
Conductor $3$
Sign $1$
Analytic cond. $35.1331$
Root an. cond. $5.92731$
Motivic weight $43$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.84e6·2-s − 1.04e10·3-s + 2.53e13·4-s + 1.12e15·5-s − 6.10e16·6-s − 1.55e18·7-s + 9.64e19·8-s + 1.09e20·9-s + 6.55e21·10-s + 2.50e22·11-s − 2.64e23·12-s + 2.47e23·13-s − 9.06e24·14-s − 1.17e25·15-s + 3.40e26·16-s − 1.03e26·17-s + 6.39e26·18-s − 6.03e26·19-s + 2.84e28·20-s + 1.62e28·21-s + 1.46e29·22-s − 3.77e28·23-s − 1.00e30·24-s + 1.22e29·25-s + 1.44e30·26-s − 1.14e30·27-s − 3.92e31·28-s + ⋯
L(s)  = 1  + 1.96·2-s − 0.577·3-s + 2.87·4-s + 1.05·5-s − 1.13·6-s − 1.05·7-s + 3.69·8-s + 0.333·9-s + 2.07·10-s + 1.02·11-s − 1.66·12-s + 0.278·13-s − 2.06·14-s − 0.607·15-s + 4.40·16-s − 0.364·17-s + 0.656·18-s − 0.194·19-s + 3.02·20-s + 0.606·21-s + 2.00·22-s − 0.199·23-s − 2.13·24-s + 0.107·25-s + 0.548·26-s − 0.192·27-s − 3.02·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(44-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+43/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $1$
Analytic conductor: \(35.1331\)
Root analytic conductor: \(5.92731\)
Motivic weight: \(43\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :43/2),\ 1)\)

Particular Values

\(L(22)\) \(\approx\) \(7.201617652\)
\(L(\frac12)\) \(\approx\) \(7.201617652\)
\(L(\frac{45}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.04e10T \)
good2 \( 1 - 5.84e6T + 8.79e12T^{2} \)
5 \( 1 - 1.12e15T + 1.13e30T^{2} \)
7 \( 1 + 1.55e18T + 2.18e36T^{2} \)
11 \( 1 - 2.50e22T + 6.02e44T^{2} \)
13 \( 1 - 2.47e23T + 7.93e47T^{2} \)
17 \( 1 + 1.03e26T + 8.11e52T^{2} \)
19 \( 1 + 6.03e26T + 9.69e54T^{2} \)
23 \( 1 + 3.77e28T + 3.58e58T^{2} \)
29 \( 1 - 4.08e31T + 7.64e62T^{2} \)
31 \( 1 - 1.22e32T + 1.34e64T^{2} \)
37 \( 1 - 1.91e33T + 2.70e67T^{2} \)
41 \( 1 + 5.34e34T + 2.23e69T^{2} \)
43 \( 1 + 3.81e33T + 1.73e70T^{2} \)
47 \( 1 + 9.23e35T + 7.94e71T^{2} \)
53 \( 1 + 7.06e36T + 1.39e74T^{2} \)
59 \( 1 + 7.34e37T + 1.40e76T^{2} \)
61 \( 1 + 4.55e38T + 5.87e76T^{2} \)
67 \( 1 - 9.19e38T + 3.32e78T^{2} \)
71 \( 1 + 1.09e40T + 4.01e79T^{2} \)
73 \( 1 - 6.59e39T + 1.32e80T^{2} \)
79 \( 1 + 1.10e41T + 3.96e81T^{2} \)
83 \( 1 + 1.89e41T + 3.31e82T^{2} \)
89 \( 1 + 1.34e41T + 6.66e83T^{2} \)
97 \( 1 - 7.60e42T + 2.69e85T^{2} \)
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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

−15.72480453218730449203495755705, −14.05297771016825676554077550617, −13.04046736872740678616415316059, −11.79990250537731085206369531976, −10.15521852414179608341851018729, −6.57600892713768839894895943071, −6.08189191622170466475631181701, −4.53928643921455225669044805885, −3.05741398354729471616315020819, −1.55444891737454959864496374162, 1.55444891737454959864496374162, 3.05741398354729471616315020819, 4.53928643921455225669044805885, 6.08189191622170466475631181701, 6.57600892713768839894895943071, 10.15521852414179608341851018729, 11.79990250537731085206369531976, 13.04046736872740678616415316059, 14.05297771016825676554077550617, 15.72480453218730449203495755705

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