Properties

Label 4-601120-1.1-c1e2-0-21
Degree $4$
Conductor $601120$
Sign $-1$
Analytic cond. $38.3279$
Root an. cond. $2.48816$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 5-s + 8-s − 4·9-s + 10-s + 3·13-s + 16-s − 4·18-s + 20-s − 4·25-s + 3·26-s − 6·29-s + 32-s − 4·36-s + 40-s − 12·41-s − 4·45-s − 14·49-s − 4·50-s + 3·52-s − 6·58-s − 18·61-s + 64-s + 3·65-s − 4·72-s + 80-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s − 4/3·9-s + 0.316·10-s + 0.832·13-s + 1/4·16-s − 0.942·18-s + 0.223·20-s − 4/5·25-s + 0.588·26-s − 1.11·29-s + 0.176·32-s − 2/3·36-s + 0.158·40-s − 1.87·41-s − 0.596·45-s − 2·49-s − 0.565·50-s + 0.416·52-s − 0.787·58-s − 2.30·61-s + 1/8·64-s + 0.372·65-s − 0.471·72-s + 0.111·80-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 601120 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 601120 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(601120\)    =    \(2^{5} \cdot 5 \cdot 13 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(38.3279\)
Root analytic conductor: \(2.48816\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 601120,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T^{2} ) \)
13$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 4 T + p T^{2} ) \)
17$C_2$ \( 1 + p T^{2} \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.11.a_ae
19$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.19.a_q
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.g_cg
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.31.a_ba
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2^2$ \( 1 + 76 T^{2} + p^{2} T^{4} \) 2.43.a_cy
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \) 2.59.a_adc
61$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.s_hu
67$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.67.a_aba
71$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \) 2.71.a_aec
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.a_fm
79$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.79.a_o
83$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.83.a_cg
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.g_ec
97$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.97.a_fa
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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

−8.195512232104151329776621363860, −7.75763148580811915441741861387, −7.25311381204152707488151928917, −6.60446915404692485943558492124, −6.24107050116923517273489837066, −5.86421870182713502291441442012, −5.50338319111461222472379830285, −4.95018083929592668407166151599, −4.49418948672783659000937386030, −3.65678147760259876170376472913, −3.37112493735724764606643990122, −2.83777844940901919462927604437, −2.01287152784650252307272933220, −1.51678282182365219880973995246, 0, 1.51678282182365219880973995246, 2.01287152784650252307272933220, 2.83777844940901919462927604437, 3.37112493735724764606643990122, 3.65678147760259876170376472913, 4.49418948672783659000937386030, 4.95018083929592668407166151599, 5.50338319111461222472379830285, 5.86421870182713502291441442012, 6.24107050116923517273489837066, 6.60446915404692485943558492124, 7.25311381204152707488151928917, 7.75763148580811915441741861387, 8.195512232104151329776621363860

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