Properties

Label 2-2e6-1.1-c19-0-6
Degree $2$
Conductor $64$
Sign $1$
Analytic cond. $146.442$
Root an. cond. $12.1013$
Motivic weight $19$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.06e4·3-s + 2.37e6·5-s − 1.69e7·7-s + 1.40e9·9-s + 1.62e7·11-s − 5.04e10·13-s − 1.20e11·15-s + 2.25e11·17-s + 1.71e12·19-s + 8.56e11·21-s + 1.40e13·23-s − 1.34e13·25-s − 1.22e13·27-s − 1.13e12·29-s − 1.04e14·31-s − 8.21e11·33-s − 4.02e13·35-s + 1.69e14·37-s + 2.55e15·39-s − 3.30e15·41-s − 1.12e15·43-s + 3.33e15·45-s + 3.49e15·47-s − 1.11e16·49-s − 1.14e16·51-s − 2.99e16·53-s + 3.85e13·55-s + ⋯
L(s)  = 1  − 1.48·3-s + 0.544·5-s − 0.158·7-s + 1.20·9-s + 0.00207·11-s − 1.31·13-s − 0.808·15-s + 0.460·17-s + 1.21·19-s + 0.235·21-s + 1.62·23-s − 0.703·25-s − 0.308·27-s − 0.0145·29-s − 0.710·31-s − 0.00308·33-s − 0.0862·35-s + 0.214·37-s + 1.95·39-s − 1.57·41-s − 0.342·43-s + 0.657·45-s + 0.456·47-s − 0.974·49-s − 0.683·51-s − 1.24·53-s + 0.00112·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $1$
Analytic conductor: \(146.442\)
Root analytic conductor: \(12.1013\)
Motivic weight: \(19\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :19/2),\ 1)\)

Particular Values

\(L(10)\) \(\approx\) \(0.9769289664\)
\(L(\frac12)\) \(\approx\) \(0.9769289664\)
\(L(\frac{21}{2})\) 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 \( 1 + 1876 p^{3} T + p^{19} T^{2} \)
5 \( 1 - 475482 p T + p^{19} T^{2} \)
7 \( 1 + 345256 p^{2} T + p^{19} T^{2} \)
11 \( 1 - 1473828 p T + p^{19} T^{2} \)
13 \( 1 + 3878585774 p T + p^{19} T^{2} \)
17 \( 1 - 13239417618 p T + p^{19} T^{2} \)
19 \( 1 - 1710278572660 T + p^{19} T^{2} \)
23 \( 1 - 14036534788872 T + p^{19} T^{2} \)
29 \( 1 + 1137835269510 T + p^{19} T^{2} \)
31 \( 1 + 104626880141728 T + p^{19} T^{2} \)
37 \( 1 - 169392327370594 T + p^{19} T^{2} \)
41 \( 1 + 3309984750560838 T + p^{19} T^{2} \)
43 \( 1 + 1127913532193492 T + p^{19} T^{2} \)
47 \( 1 - 3498693987674256 T + p^{19} T^{2} \)
53 \( 1 + 29956294112980302 T + p^{19} T^{2} \)
59 \( 1 + 58391397642732420 T + p^{19} T^{2} \)
61 \( 1 + 23373685132672742 T + p^{19} T^{2} \)
67 \( 1 - 205102524257382244 T + p^{19} T^{2} \)
71 \( 1 + 177902341950417768 T + p^{19} T^{2} \)
73 \( 1 - 299853775038660122 T + p^{19} T^{2} \)
79 \( 1 + 92227090144007440 T + p^{19} T^{2} \)
83 \( 1 + 1208542823470585932 T + p^{19} T^{2} \)
89 \( 1 - 4371201192290304330 T + p^{19} T^{2} \)
97 \( 1 + 635013222218448094 T + p^{19} T^{2} \)
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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

−11.30139658906966558257206209594, −10.18538137082687661977453391286, −9.387688526107799403590231283361, −7.52510229546758450926298979142, −6.56255033234105818320739888536, −5.42479685274641551901673366227, −4.88500668343664330359261361719, −3.12296501732104396824758888180, −1.60707189225044031553948379450, −0.48858854388928965352795133584, 0.48858854388928965352795133584, 1.60707189225044031553948379450, 3.12296501732104396824758888180, 4.88500668343664330359261361719, 5.42479685274641551901673366227, 6.56255033234105818320739888536, 7.52510229546758450926298979142, 9.387688526107799403590231283361, 10.18538137082687661977453391286, 11.30139658906966558257206209594

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