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Properties

Label 4-390e2-1.1-c1e2-0-22
Degree $4$
Conductor $152100$
Sign $1$
Analytic cond. $9.69802$
Root an. cond. $1.76469$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 4·5-s − 9-s − 12·11-s + 16-s − 12·19-s + 4·20-s + 11·25-s − 4·29-s + 8·31-s + 36-s − 12·41-s + 12·44-s + 4·45-s + 14·49-s + 48·55-s − 20·59-s − 12·61-s − 64-s − 16·71-s + 12·76-s − 32·79-s − 4·80-s + 81-s + 20·89-s + 48·95-s + 12·99-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.78·5-s − 1/3·9-s − 3.61·11-s + 1/4·16-s − 2.75·19-s + 0.894·20-s + 11/5·25-s − 0.742·29-s + 1.43·31-s + 1/6·36-s − 1.87·41-s + 1.80·44-s + 0.596·45-s + 2·49-s + 6.47·55-s − 2.60·59-s − 1.53·61-s − 1/8·64-s − 1.89·71-s + 1.37·76-s − 3.60·79-s − 0.447·80-s + 1/9·81-s + 2.11·89-s + 4.92·95-s + 1.20·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(152100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(9.69802\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 152100,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
13$C_2$ \( 1 + T^{2} \)
good7$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.7.a_ao
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.11.m_cg
17$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.17.a_abi
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.19.m_cw
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.37.a_ba
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.43.a_aw
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.47.a_abe
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.53.a_acs
59$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.59.u_ik
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.61.m_gc
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.67.a_aeo
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.71.q_hy
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.a_aeg
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.79.bg_py
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \) 2.83.a_afu
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \) 2.97.a_ahi
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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

−10.85240569269252447258421599262, −10.70809765424301575703797741022, −10.27860822179830146277111877190, −10.09877033016297426394059805825, −8.834150179522979669111286677836, −8.721114826406882882491544725395, −8.262799860054498916787014023187, −7.914469002022591089368447615768, −7.47020225606372979519662389201, −7.16534286090752305271621679780, −6.15728573959655977856396757472, −5.79275595864651550240912883168, −4.98628006169101036492933345719, −4.55528528910644154233824829377, −4.36069336101473660250875034031, −3.28416136422331593769413605176, −2.90566405700736196445289808821, −2.18171372071285739740026906105, 0, 0, 2.18171372071285739740026906105, 2.90566405700736196445289808821, 3.28416136422331593769413605176, 4.36069336101473660250875034031, 4.55528528910644154233824829377, 4.98628006169101036492933345719, 5.79275595864651550240912883168, 6.15728573959655977856396757472, 7.16534286090752305271621679780, 7.47020225606372979519662389201, 7.914469002022591089368447615768, 8.262799860054498916787014023187, 8.721114826406882882491544725395, 8.834150179522979669111286677836, 10.09877033016297426394059805825, 10.27860822179830146277111877190, 10.70809765424301575703797741022, 10.85240569269252447258421599262

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