Properties

Label 4-1064960-1.1-c1e2-0-3
Degree $4$
Conductor $1064960$
Sign $1$
Analytic cond. $67.9027$
Root an. cond. $2.87059$
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  − 5-s − 4·9-s + 5·13-s + 6·17-s − 4·25-s − 6·29-s + 4·45-s + 14·49-s + 12·53-s − 18·61-s − 5·65-s − 4·73-s + 7·81-s − 6·85-s − 6·89-s − 16·97-s + 12·101-s + 24·113-s − 20·117-s + 4·121-s + 4·125-s + ⋯
L(s)  = 1  − 0.447·5-s − 4/3·9-s + 1.38·13-s + 1.45·17-s − 4/5·25-s − 1.11·29-s + 0.596·45-s + 2·49-s + 1.64·53-s − 2.30·61-s − 0.620·65-s − 0.468·73-s + 7/9·81-s − 0.650·85-s − 0.635·89-s − 1.62·97-s + 1.19·101-s + 2.25·113-s − 1.84·117-s + 4/11·121-s + 0.357·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1064960 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1064960 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1064960\)    =    \(2^{14} \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(67.9027\)
Root analytic conductor: \(2.87059\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1064960,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.528506326\)
\(L(\frac12)\) \(\approx\) \(1.528506326\)
\(L(\frac{3}{2})\) 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$ \( ( 1 - T )( 1 - 4 T + p T^{2} ) \)
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_ao
11$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.11.a_ae
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.17.ag_bi
19$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.19.a_aq
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.g_cg
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.31.a_aba
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2^2$ \( 1 - 76 T^{2} + p^{2} T^{4} \) 2.43.a_acy
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \) 2.59.a_dc
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_ba
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.a_ec
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.79.a_ao
83$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.83.a_acg
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.g_ec
97$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.97.q_jy
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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

−8.125786976803434621813973215205, −7.69362039401877980804081114216, −7.30958101174655155533886852167, −6.86729322932774074254153541198, −6.04259145738153328001269293535, −5.78428844075377618172334748131, −5.71171892910742761721693602323, −5.05160892174405862318170039275, −4.24910273419342915155643544636, −3.91137618976364794283231896479, −3.33722256615425598832314112054, −3.02924838323870685688688470358, −2.23918987059996248177706257688, −1.46285011639832764534570322432, −0.58706007474067901828046223574, 0.58706007474067901828046223574, 1.46285011639832764534570322432, 2.23918987059996248177706257688, 3.02924838323870685688688470358, 3.33722256615425598832314112054, 3.91137618976364794283231896479, 4.24910273419342915155643544636, 5.05160892174405862318170039275, 5.71171892910742761721693602323, 5.78428844075377618172334748131, 6.04259145738153328001269293535, 6.86729322932774074254153541198, 7.30958101174655155533886852167, 7.69362039401877980804081114216, 8.125786976803434621813973215205

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