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Properties

Label 4-266240-1.1-c1e2-0-16
Degree $4$
Conductor $266240$
Sign $-1$
Analytic cond. $16.9756$
Root an. cond. $2.02981$
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  + 5-s − 4·9-s − 13-s + 2·17-s − 4·25-s − 6·29-s + 8·37-s − 8·41-s − 4·45-s − 2·49-s + 8·53-s − 6·61-s − 65-s + 7·81-s + 2·85-s − 14·89-s − 4·97-s + 8·101-s + 8·109-s − 24·113-s + 4·117-s − 8·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 0.447·5-s − 4/3·9-s − 0.277·13-s + 0.485·17-s − 4/5·25-s − 1.11·29-s + 1.31·37-s − 1.24·41-s − 0.596·45-s − 2/7·49-s + 1.09·53-s − 0.768·61-s − 0.124·65-s + 7/9·81-s + 0.216·85-s − 1.48·89-s − 0.406·97-s + 0.796·101-s + 0.766·109-s − 2.25·113-s + 0.369·117-s − 0.727·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 266240 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266240 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(266240\)    =    \(2^{12} \cdot 5 \cdot 13\)
Sign: $-1$
Analytic conductor: \(16.9756\)
Root analytic conductor: \(2.02981\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 266240,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T^{2} ) \)
13$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.11.a_i
17$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.ac_ba
19$C_2^2$ \( 1 - 28 T^{2} + p^{2} T^{4} \) 2.19.a_abc
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.23.a_c
29$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.g_co
31$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.31.a_abi
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ai_cc
41$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.i_dq
43$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \) 2.43.a_bs
47$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.47.a_abi
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.53.ai_eo
59$C_2^2$ \( 1 + 100 T^{2} + p^{2} T^{4} \) 2.59.a_dw
61$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.61.g_es
67$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \) 2.67.a_co
71$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.71.a_fi
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.a_fm
79$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \) 2.79.a_co
83$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.83.a_cs
89$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.89.o_fq
97$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.97.e_gg
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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

−8.648925138258895552227932332077, −8.275035483699726333748069982551, −7.70423574508221566994927859461, −7.40412874430029907030366210036, −6.69121020649730915771536699811, −6.14889242333298511061408411220, −5.79288960691191945788133270996, −5.33806468742439573688935176545, −4.87931202647567556511001257932, −4.03361949953061797713669675462, −3.55124661221935437990644631764, −2.79628907618626014293070762860, −2.33050880860435460577428186862, −1.41374986956311577260790484896, 0, 1.41374986956311577260790484896, 2.33050880860435460577428186862, 2.79628907618626014293070762860, 3.55124661221935437990644631764, 4.03361949953061797713669675462, 4.87931202647567556511001257932, 5.33806468742439573688935176545, 5.79288960691191945788133270996, 6.14889242333298511061408411220, 6.69121020649730915771536699811, 7.40412874430029907030366210036, 7.70423574508221566994927859461, 8.275035483699726333748069982551, 8.648925138258895552227932332077

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