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Properties

Label 4-3300e2-1.1-c1e2-0-12
Degree $4$
Conductor $10890000$
Sign $1$
Analytic cond. $694.355$
Root an. cond. $5.13328$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 9-s + 2·11-s + 12·19-s + 16·29-s − 16·31-s + 16·41-s + 10·49-s − 24·59-s + 20·61-s + 16·71-s + 4·79-s + 81-s + 28·89-s − 2·99-s − 32·101-s + 20·109-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + ⋯
L(s)  = 1  − 1/3·9-s + 0.603·11-s + 2.75·19-s + 2.97·29-s − 2.87·31-s + 2.49·41-s + 10/7·49-s − 3.12·59-s + 2.56·61-s + 1.89·71-s + 0.450·79-s + 1/9·81-s + 2.96·89-s − 0.201·99-s − 3.18·101-s + 1.91·109-s + 3/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10890000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10890000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10890000\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(694.355\)
Root analytic conductor: \(5.13328\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10890000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.697222790\)
\(L(\frac12)\) \(\approx\) \(3.697222790\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.19.am_cw
23$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.23.a_abu
29$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.29.aq_es
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.31.q_ew
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.37.a_ba
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.41.aq_fq
43$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.43.a_ade
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.47.a_abe
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.53.a_ady
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.59.y_kc
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.61.au_io
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.67.a_k
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.a_aeg
79$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.79.ae_gg
83$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \) 2.83.a_dm
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.89.abc_ok
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

−8.964543476441895619854935788781, −8.494913972075969112483859501835, −7.918483050373540834725268828865, −7.74605146323033103501731218108, −7.33282486254383327967625189608, −7.09959620284081841832896129480, −6.47138926975835557601854511766, −6.31495171883503283910924525880, −5.66579187298214255086375364133, −5.47415499105600653095977112259, −5.04860348054364907420789976854, −4.69739602723093305136970329639, −3.97145991633301667778086157478, −3.84946862447953932051315066287, −3.10548905861548274337659435618, −3.00077528210424142678170074750, −2.33464781868136450376704728862, −1.76478075288825487983398612164, −0.899771228228213567001215918723, −0.812736479994241797450003334490, 0.812736479994241797450003334490, 0.899771228228213567001215918723, 1.76478075288825487983398612164, 2.33464781868136450376704728862, 3.00077528210424142678170074750, 3.10548905861548274337659435618, 3.84946862447953932051315066287, 3.97145991633301667778086157478, 4.69739602723093305136970329639, 5.04860348054364907420789976854, 5.47415499105600653095977112259, 5.66579187298214255086375364133, 6.31495171883503283910924525880, 6.47138926975835557601854511766, 7.09959620284081841832896129480, 7.33282486254383327967625189608, 7.74605146323033103501731218108, 7.918483050373540834725268828865, 8.494913972075969112483859501835, 8.964543476441895619854935788781

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