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Properties

Label 4-20608-1.1-c1e2-0-2
Degree $4$
Conductor $20608$
Sign $1$
Analytic cond. $1.31398$
Root an. cond. $1.07064$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 3·7-s − 8-s + 4·9-s − 3·14-s + 16-s − 4·18-s − 5·23-s + 2·25-s + 3·28-s − 8·31-s − 32-s + 4·36-s + 5·46-s + 6·47-s − 2·50-s − 3·56-s + 8·62-s + 12·63-s + 64-s − 18·71-s − 4·72-s − 2·73-s + 16·79-s + 7·81-s − 12·89-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 1.13·7-s − 0.353·8-s + 4/3·9-s − 0.801·14-s + 1/4·16-s − 0.942·18-s − 1.04·23-s + 2/5·25-s + 0.566·28-s − 1.43·31-s − 0.176·32-s + 2/3·36-s + 0.737·46-s + 0.875·47-s − 0.282·50-s − 0.400·56-s + 1.01·62-s + 1.51·63-s + 1/8·64-s − 2.13·71-s − 0.471·72-s − 0.234·73-s + 1.80·79-s + 7/9·81-s − 1.27·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20608 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20608 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(20608\)    =    \(2^{7} \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(1.31398\)
Root analytic conductor: \(1.07064\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 20608,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.021023512\)
\(L(\frac12)\) \(\approx\) \(1.021023512\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2$C_1$ \( 1 + T \)
7$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
23$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 6 T + p T^{2} ) \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.3.a_ae
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
11$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.11.a_ai
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.13.a_ao
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
29$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.29.a_k
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.37.a_bi
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \) 2.43.a_cm
47$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.47.ag_dq
53$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.53.a_k
59$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \) 2.59.a_abg
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.a_cg
67$C_2^2$ \( 1 - 68 T^{2} + p^{2} T^{4} \) 2.67.a_acq
71$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.s_ig
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.73.c_fi
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.a_fa
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.97.aq_io
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.77391838373595355131522186042, −10.23997581603767042876261615702, −9.908003354304607301192635825633, −9.138707492896557792353954076963, −8.764524696088721026224024358157, −8.060807845904275559983198993142, −7.49278481429302024443317855098, −7.27458725908727650827385778875, −6.43387335364160770790015500871, −5.75081849682282132287250666301, −4.94274876815638520101943605736, −4.32865884310402845291045347589, −3.54990991519467242994116346372, −2.20620857364983596953646873319, −1.43439069980046604356623977304, 1.43439069980046604356623977304, 2.20620857364983596953646873319, 3.54990991519467242994116346372, 4.32865884310402845291045347589, 4.94274876815638520101943605736, 5.75081849682282132287250666301, 6.43387335364160770790015500871, 7.27458725908727650827385778875, 7.49278481429302024443317855098, 8.060807845904275559983198993142, 8.764524696088721026224024358157, 9.138707492896557792353954076963, 9.908003354304607301192635825633, 10.23997581603767042876261615702, 10.77391838373595355131522186042

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