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Properties

Label 4-770e2-1.1-c1e2-0-13
Degree $4$
Conductor $592900$
Sign $1$
Analytic cond. $37.8038$
Root an. cond. $2.47961$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s + 3·7-s + 4·8-s − 5·9-s + 2·11-s + 6·14-s + 5·16-s − 10·18-s + 4·22-s − 12·23-s + 25-s + 9·28-s + 10·29-s + 6·32-s − 15·36-s + 6·37-s + 8·43-s + 6·44-s − 24·46-s + 2·49-s + 2·50-s − 2·53-s + 12·56-s + 20·58-s − 15·63-s + 7·64-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.13·7-s + 1.41·8-s − 5/3·9-s + 0.603·11-s + 1.60·14-s + 5/4·16-s − 2.35·18-s + 0.852·22-s − 2.50·23-s + 1/5·25-s + 1.70·28-s + 1.85·29-s + 1.06·32-s − 5/2·36-s + 0.986·37-s + 1.21·43-s + 0.904·44-s − 3.53·46-s + 2/7·49-s + 0.282·50-s − 0.274·53-s + 1.60·56-s + 2.62·58-s − 1.88·63-s + 7/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(592900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(37.8038\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 592900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.201342520\)
\(L(\frac12)\) \(\approx\) \(5.201342520\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_2$ \( 1 - 3 T + p T^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.3.a_f
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.17.a_ap
19$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.19.a_n
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.23.m_de
29$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.29.ak_df
31$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.31.a_cb
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.37.ag_df
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.a_da
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.47.a_dm
53$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.53.c_ed
59$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.59.a_s
61$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.61.a_cv
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.67.aq_hq
71$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.71.ao_hj
73$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.a_aby
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.79.au_jy
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.a_fa
89$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.89.a_abv
97$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.97.a_by
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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

−8.179100717490198570895041997468, −8.040279775818558509679448913672, −7.64414702723538290730003684901, −6.71133528252224918895941305968, −6.45727768735845281382720038316, −6.09792541147733225681149376805, −5.43280032020194587138281999658, −5.32819444768541291953052917923, −4.57429547993073909832853594043, −4.22394490747904301331264691816, −3.70218143630793948520871859223, −3.07617612477761288858296720437, −2.34744313488402780667751356221, −2.10347249498215262240223155354, −0.948761287194466744858706658557, 0.948761287194466744858706658557, 2.10347249498215262240223155354, 2.34744313488402780667751356221, 3.07617612477761288858296720437, 3.70218143630793948520871859223, 4.22394490747904301331264691816, 4.57429547993073909832853594043, 5.32819444768541291953052917923, 5.43280032020194587138281999658, 6.09792541147733225681149376805, 6.45727768735845281382720038316, 6.71133528252224918895941305968, 7.64414702723538290730003684901, 8.040279775818558509679448913672, 8.179100717490198570895041997468

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