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Properties

Label 4-2700e2-1.1-c1e2-0-6
Degree $4$
Conductor $7290000$
Sign $1$
Analytic cond. $464.816$
Root an. cond. $4.64323$
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  + 2·19-s + 22·31-s − 11·49-s − 26·61-s − 34·79-s − 34·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  + 0.458·19-s + 3.95·31-s − 1.57·49-s − 3.32·61-s − 3.82·79-s − 3.25·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7290000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7290000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7290000\)    =    \(2^{4} \cdot 3^{6} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(464.816\)
Root analytic conductor: \(4.64323\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7290000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.017205262\)
\(L(\frac12)\) \(\approx\) \(2.017205262\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \) 2.7.a_l
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.17.a_abi
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.19.ac_bn
23$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.23.a_abu
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.31.aw_hb
37$C_2^2$ \( 1 + 47 T^{2} + p^{2} T^{4} \) 2.37.a_bv
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2^2$ \( 1 + 83 T^{2} + p^{2} T^{4} \) 2.43.a_df
47$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.47.a_adq
53$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.53.a_aec
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.61.ba_lf
67$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \) 2.67.a_es
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \) 2.73.a_adt
79$C_2$ \( ( 1 + 17 T + p T^{2} )^{2} \) 2.79.bi_rf
83$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.83.a_agk
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2^2$ \( 1 - 169 T^{2} + p^{2} T^{4} \) 2.97.a_agn
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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

−9.085934589697565499931874923028, −8.554375031386572192766114440581, −8.249257195434094560268135290200, −7.895836742145397942675674466628, −7.66345477838261376773631625211, −7.13720056262630286276754617072, −6.49882639138016679980529514382, −6.49735603104344151592564699896, −6.10842584000220166381384788243, −5.46372097465066060297320043637, −5.19333582488680583501874918662, −4.58202955010652334834121685459, −4.35109969994434934622480930644, −4.02292273172673601132911816275, −3.07952888103989891181845320424, −2.91461285961539964307065596914, −2.66986118010616510334576986238, −1.56702030443397789125754367025, −1.40234944128730670629850968073, −0.47881842568646752659034258515, 0.47881842568646752659034258515, 1.40234944128730670629850968073, 1.56702030443397789125754367025, 2.66986118010616510334576986238, 2.91461285961539964307065596914, 3.07952888103989891181845320424, 4.02292273172673601132911816275, 4.35109969994434934622480930644, 4.58202955010652334834121685459, 5.19333582488680583501874918662, 5.46372097465066060297320043637, 6.10842584000220166381384788243, 6.49735603104344151592564699896, 6.49882639138016679980529514382, 7.13720056262630286276754617072, 7.66345477838261376773631625211, 7.895836742145397942675674466628, 8.249257195434094560268135290200, 8.554375031386572192766114440581, 9.085934589697565499931874923028

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