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Properties

Label 4-1155e2-1.1-c1e2-0-15
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·4-s + 2·5-s + 9-s − 4·11-s + 5·16-s + 6·20-s − 25-s + 3·36-s − 12·44-s + 2·45-s − 49-s − 8·55-s + 24·59-s + 3·64-s + 10·80-s + 81-s + 28·89-s − 4·99-s − 3·100-s + 5·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 5·144-s + 149-s + ⋯
L(s)  = 1  + 3/2·4-s + 0.894·5-s + 1/3·9-s − 1.20·11-s + 5/4·16-s + 1.34·20-s − 1/5·25-s + 1/2·36-s − 1.80·44-s + 0.298·45-s − 1/7·49-s − 1.07·55-s + 3.12·59-s + 3/8·64-s + 1.11·80-s + 1/9·81-s + 2.96·89-s − 0.402·99-s − 0.299·100-s + 5/11·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5/12·144-s + 0.0819·149-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: \(0\)
Selberg data: \((4,\ 1334025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.719757743\)
\(L(\frac12)\) \(\approx\) \(3.719757743\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
11$C_2$ \( 1 + 4 T + p T^{2} \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.2.a_ad
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.a_cc
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.a_bm
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.a_da
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.43.a_acs
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.59.ay_kc
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.a_eo
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.a_aeg
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.79.a_adu
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.83.a_aw
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.89.abc_ok
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.a_afa
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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

−7.82457535345750901578753047461, −7.60623941244305847293816632860, −6.96363218536198676673386336462, −6.68389782671828476586716068601, −6.35142955422321351185834919937, −5.72477212889125210848152415284, −5.47512304741807642262739225829, −5.07201920078771283162396189203, −4.37554171346669705530079751011, −3.74729235826333790072734794366, −3.12075659632733233853047705193, −2.62794985537489172732744210939, −2.08480612512781162744456009386, −1.84267915178081397005158374246, −0.810080635104918364365676292524, 0.810080635104918364365676292524, 1.84267915178081397005158374246, 2.08480612512781162744456009386, 2.62794985537489172732744210939, 3.12075659632733233853047705193, 3.74729235826333790072734794366, 4.37554171346669705530079751011, 5.07201920078771283162396189203, 5.47512304741807642262739225829, 5.72477212889125210848152415284, 6.35142955422321351185834919937, 6.68389782671828476586716068601, 6.96363218536198676673386336462, 7.60623941244305847293816632860, 7.82457535345750901578753047461

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