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Properties

Label 4-1155e2-1.1-c1e2-0-33
Degree $4$
Conductor $1334025$
Sign $1$
Analytic cond. $85.0585$
Root an. cond. $3.03689$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·4-s + 3·9-s − 11-s + 4·12-s − 2·23-s − 5·25-s − 4·27-s − 14·31-s + 2·33-s − 6·36-s + 5·37-s + 2·44-s − 16·47-s − 49-s − 8·53-s − 18·59-s + 8·64-s + 67-s + 4·69-s − 25·71-s + 10·75-s + 5·81-s + 4·89-s + 4·92-s + 28·93-s − 10·97-s + ⋯
L(s)  = 1  − 1.15·3-s − 4-s + 9-s − 0.301·11-s + 1.15·12-s − 0.417·23-s − 25-s − 0.769·27-s − 2.51·31-s + 0.348·33-s − 36-s + 0.821·37-s + 0.301·44-s − 2.33·47-s − 1/7·49-s − 1.09·53-s − 2.34·59-s + 64-s + 0.122·67-s + 0.481·69-s − 2.96·71-s + 1.15·75-s + 5/9·81-s + 0.423·89-s + 0.417·92-s + 2.90·93-s − 1.01·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1334025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1334025\)    =    \(3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(85.0585\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1334025,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad3$C_1$ \( ( 1 + T )^{2} \)
5$C_2$ \( 1 + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
11$C_2$ \( 1 + T + p T^{2} \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.2.a_c
13$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.13.a_af
17$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.17.a_ad
19$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.19.a_az
23$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.23.c_br
29$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.29.a_an
31$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.31.o_dy
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.37.af_cq
41$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.41.a_ad
43$C_2^2$ \( 1 + 51 T^{2} + p^{2} T^{4} \) 2.43.a_bz
47$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.47.q_fy
53$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.53.i_cv
59$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.s_hi
61$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \) 2.61.a_acn
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.67.ab_es
71$C_2$$\times$$C_2$ \( ( 1 + 9 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.71.z_la
73$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.73.a_az
79$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \) 2.79.a_acn
83$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.83.a_ack
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.ae_eo
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.97.k_go
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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

−7.66735462906469826139046757966, −7.13452436940323001646950817878, −6.47423600077163054216956604514, −6.19327534981179256142318579830, −5.73576747452596683079961624564, −5.28670753225953059408128050340, −4.88865266423701574708865806266, −4.47133408764934192731976235160, −4.02463357393593844423485227769, −3.50743393785278027900437917571, −2.86753379578947281936970614352, −1.86519265599361460087835495711, −1.44852490534330317972811698973, 0, 0, 1.44852490534330317972811698973, 1.86519265599361460087835495711, 2.86753379578947281936970614352, 3.50743393785278027900437917571, 4.02463357393593844423485227769, 4.47133408764934192731976235160, 4.88865266423701574708865806266, 5.28670753225953059408128050340, 5.73576747452596683079961624564, 6.19327534981179256142318579830, 6.47423600077163054216956604514, 7.13452436940323001646950817878, 7.66735462906469826139046757966

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