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Properties

Label 4-132300-1.1-c1e2-0-30
Degree $4$
Conductor $132300$
Sign $-1$
Analytic cond. $8.43556$
Root an. cond. $1.70423$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 4-s + 9-s − 12-s − 4·13-s + 16-s − 4·19-s − 25-s + 27-s − 12·31-s − 36-s − 12·37-s − 4·39-s − 8·43-s + 48-s + 49-s + 4·52-s − 4·57-s + 4·61-s − 64-s − 8·67-s + 4·73-s − 75-s + 4·76-s + 8·79-s + 81-s − 12·93-s + ⋯
L(s)  = 1  + 0.577·3-s − 1/2·4-s + 1/3·9-s − 0.288·12-s − 1.10·13-s + 1/4·16-s − 0.917·19-s − 1/5·25-s + 0.192·27-s − 2.15·31-s − 1/6·36-s − 1.97·37-s − 0.640·39-s − 1.21·43-s + 0.144·48-s + 1/7·49-s + 0.554·52-s − 0.529·57-s + 0.512·61-s − 1/8·64-s − 0.977·67-s + 0.468·73-s − 0.115·75-s + 0.458·76-s + 0.900·79-s + 1/9·81-s − 1.24·93-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132300 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132300 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(132300\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(8.43556\)
Root analytic conductor: \(1.70423\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 132300,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
5$C_2$ \( 1 + T^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.e_ba
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.17.a_ao
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.19.e_bq
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.23.a_s
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.29.a_ak
31$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.31.m_de
37$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.m_dq
41$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.41.a_bi
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.47.a_abi
53$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.53.a_aw
59$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.59.a_acw
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.61.ae_ck
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.i_di
71$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.71.a_as
73$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.73.ae_fq
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.79.ai_gs
83$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \) 2.83.a_aec
89$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \) 2.89.a_aeg
97$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.97.au_le
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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

−9.047766904033734993611769964802, −8.785744392017225320829519701952, −8.191758363650486041245683368094, −7.72503893242390875989055799913, −7.18431918002909293426474519824, −6.82469588581458165251268679134, −6.14907276799980263712284982121, −5.34738577776015290229101564262, −5.07934364300790700464068559829, −4.39313305249993442518944093890, −3.72152379611834110759014672741, −3.28821431615447895745740402384, −2.30173733562722089365856873889, −1.73024470896747153755563640960, 0, 1.73024470896747153755563640960, 2.30173733562722089365856873889, 3.28821431615447895745740402384, 3.72152379611834110759014672741, 4.39313305249993442518944093890, 5.07934364300790700464068559829, 5.34738577776015290229101564262, 6.14907276799980263712284982121, 6.82469588581458165251268679134, 7.18431918002909293426474519824, 7.72503893242390875989055799913, 8.191758363650486041245683368094, 8.785744392017225320829519701952, 9.047766904033734993611769964802

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