Properties

Label 2-2e6-1.1-c23-0-36
Degree $2$
Conductor $64$
Sign $-1$
Analytic cond. $214.530$
Root an. cond. $14.6468$
Motivic weight $23$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.11e5·3-s + 6.64e7·5-s + 8.09e9·7-s + 1.67e11·9-s + 1.78e11·11-s + 1.22e13·13-s − 3.39e13·15-s − 1.77e14·17-s + 7.96e14·19-s − 4.14e15·21-s − 2.89e15·23-s − 7.50e15·25-s − 3.77e16·27-s − 4.86e16·29-s − 1.32e17·31-s − 9.12e16·33-s + 5.37e17·35-s − 5.80e17·37-s − 6.27e18·39-s − 2.82e18·41-s + 1.78e18·43-s + 1.11e19·45-s − 1.39e19·47-s + 3.82e19·49-s + 9.06e19·51-s − 1.07e20·53-s + 1.18e19·55-s + ⋯
L(s)  = 1  − 1.66·3-s + 0.608·5-s + 1.54·7-s + 1.78·9-s + 0.188·11-s + 1.89·13-s − 1.01·15-s − 1.25·17-s + 1.56·19-s − 2.58·21-s − 0.632·23-s − 0.629·25-s − 1.30·27-s − 0.739·29-s − 0.937·31-s − 0.314·33-s + 0.941·35-s − 0.536·37-s − 3.16·39-s − 0.802·41-s + 0.292·43-s + 1.08·45-s − 0.820·47-s + 1.39·49-s + 2.09·51-s − 1.59·53-s + 0.114·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $-1$
Analytic conductor: \(214.530\)
Root analytic conductor: \(14.6468\)
Motivic weight: \(23\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 64,\ (\ :23/2),\ -1)\)

Particular Values

\(L(12)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{25}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + 5.11e5T + 9.41e10T^{2} \)
5 \( 1 - 6.64e7T + 1.19e16T^{2} \)
7 \( 1 - 8.09e9T + 2.73e19T^{2} \)
11 \( 1 - 1.78e11T + 8.95e23T^{2} \)
13 \( 1 - 1.22e13T + 4.17e25T^{2} \)
17 \( 1 + 1.77e14T + 1.99e28T^{2} \)
19 \( 1 - 7.96e14T + 2.57e29T^{2} \)
23 \( 1 + 2.89e15T + 2.08e31T^{2} \)
29 \( 1 + 4.86e16T + 4.31e33T^{2} \)
31 \( 1 + 1.32e17T + 2.00e34T^{2} \)
37 \( 1 + 5.80e17T + 1.17e36T^{2} \)
41 \( 1 + 2.82e18T + 1.24e37T^{2} \)
43 \( 1 - 1.78e18T + 3.71e37T^{2} \)
47 \( 1 + 1.39e19T + 2.87e38T^{2} \)
53 \( 1 + 1.07e20T + 4.55e39T^{2} \)
59 \( 1 - 5.27e19T + 5.36e40T^{2} \)
61 \( 1 + 1.12e20T + 1.15e41T^{2} \)
67 \( 1 + 1.35e21T + 9.99e41T^{2} \)
71 \( 1 - 4.67e20T + 3.79e42T^{2} \)
73 \( 1 - 2.11e20T + 7.18e42T^{2} \)
79 \( 1 - 1.15e21T + 4.42e43T^{2} \)
83 \( 1 - 1.97e22T + 1.37e44T^{2} \)
89 \( 1 + 4.49e22T + 6.85e44T^{2} \)
97 \( 1 - 7.68e22T + 4.96e45T^{2} \)
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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

−10.57572472126345712988224709377, −9.159745765896552521164332725584, −7.84376558425581498562695535466, −6.52767489174579572205820927696, −5.68857251563522981631332257453, −4.97608933320919175985850488180, −3.85760554764912384823317051291, −1.70233288118952416423590996189, −1.31187778212415735071759100028, 0, 1.31187778212415735071759100028, 1.70233288118952416423590996189, 3.85760554764912384823317051291, 4.97608933320919175985850488180, 5.68857251563522981631332257453, 6.52767489174579572205820927696, 7.84376558425581498562695535466, 9.159745765896552521164332725584, 10.57572472126345712988224709377

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