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Properties

Label 4-9680e2-1.1-c1e2-0-16
Degree $4$
Conductor $93702400$
Sign $1$
Analytic cond. $5974.54$
Root an. cond. $8.79176$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 2·5-s − 3·9-s + 4·15-s + 3·25-s − 14·27-s + 4·31-s − 16·37-s − 6·45-s + 6·47-s − 11·49-s − 24·53-s + 12·59-s − 14·67-s − 12·71-s + 6·75-s − 4·81-s − 18·89-s + 8·93-s − 8·97-s + 32·103-s − 32·111-s − 12·113-s + 4·125-s + 127-s + 131-s − 28·135-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.894·5-s − 9-s + 1.03·15-s + 3/5·25-s − 2.69·27-s + 0.718·31-s − 2.63·37-s − 0.894·45-s + 0.875·47-s − 1.57·49-s − 3.29·53-s + 1.56·59-s − 1.71·67-s − 1.42·71-s + 0.692·75-s − 4/9·81-s − 1.90·89-s + 0.829·93-s − 0.812·97-s + 3.15·103-s − 3.03·111-s − 1.12·113-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s − 2.40·135-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93702400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93702400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(93702400\)    =    \(2^{8} \cdot 5^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(5974.54\)
Root analytic conductor: \(8.79176\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 93702400,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 - T )^{2} \)
11 \( 1 \)
good3$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.3.ac_h
7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \) 2.7.a_l
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.19.a_ak
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.31.ae_co
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.37.q_fi
41$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \) 2.41.a_cd
43$C_2^2$ \( 1 + 59 T^{2} + p^{2} T^{4} \) 2.43.a_ch
47$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.47.ag_dz
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.53.y_jq
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.59.am_fy
61$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.61.a_az
67$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.67.o_hb
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.71.m_gw
73$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \) 2.73.a_du
79$C_2^2$ \( 1 + 146 T^{2} + p^{2} T^{4} \) 2.79.a_fq
83$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.83.a_cg
89$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.89.s_jz
97$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.97.i_ic
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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

−7.53425677247202445915154930283, −7.30153163245751144285916004001, −6.73955568369899846338690720145, −6.54483997956769996864119195578, −6.05756029874302393469627580298, −5.90421539650305055435161749208, −5.39957389208675059275610494019, −5.25930983228565534409246039580, −4.61564492177599448967714947776, −4.56002362830400627900733425879, −3.79852086146731192522019667812, −3.34173877907920924648656678680, −3.26539495356181322856977785868, −2.87653108983217856723446827956, −2.28025381849714230302570465060, −2.22544394005948352741040083431, −1.45890271118594015749355184423, −1.33518402125439306153374988995, 0, 0, 1.33518402125439306153374988995, 1.45890271118594015749355184423, 2.22544394005948352741040083431, 2.28025381849714230302570465060, 2.87653108983217856723446827956, 3.26539495356181322856977785868, 3.34173877907920924648656678680, 3.79852086146731192522019667812, 4.56002362830400627900733425879, 4.61564492177599448967714947776, 5.25930983228565534409246039580, 5.39957389208675059275610494019, 5.90421539650305055435161749208, 6.05756029874302393469627580298, 6.54483997956769996864119195578, 6.73955568369899846338690720145, 7.30153163245751144285916004001, 7.53425677247202445915154930283

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