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Properties

Label 4-3312e2-1.1-c1e2-0-9
Degree $4$
Conductor $10969344$
Sign $1$
Analytic cond. $699.414$
Root an. cond. $5.14261$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 2·7-s + 4·11-s − 4·13-s − 2·17-s + 6·19-s + 2·23-s − 4·25-s + 4·29-s + 12·31-s + 4·35-s − 8·37-s + 2·43-s + 8·47-s − 8·49-s + 18·53-s + 8·55-s − 8·59-s − 8·61-s − 8·65-s + 2·67-s + 8·71-s − 12·73-s + 8·77-s − 6·79-s + 12·83-s − 4·85-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.755·7-s + 1.20·11-s − 1.10·13-s − 0.485·17-s + 1.37·19-s + 0.417·23-s − 4/5·25-s + 0.742·29-s + 2.15·31-s + 0.676·35-s − 1.31·37-s + 0.304·43-s + 1.16·47-s − 8/7·49-s + 2.47·53-s + 1.07·55-s − 1.04·59-s − 1.02·61-s − 0.992·65-s + 0.244·67-s + 0.949·71-s − 1.40·73-s + 0.911·77-s − 0.675·79-s + 1.31·83-s − 0.433·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10969344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10969344 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10969344\)    =    \(2^{8} \cdot 3^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(699.414\)
Root analytic conductor: \(5.14261\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10969344,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.172721813\)
\(L(\frac12)\) \(\approx\) \(4.172721813\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23$C_1$ \( ( 1 - T )^{2} \)
good5$D_{4}$ \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.5.ac_i
7$D_{4}$ \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_m
11$D_{4}$ \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.11.ae_o
13$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.13.e_s
17$D_{4}$ \( 1 + 2 T + 32 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.17.c_bg
19$D_{4}$ \( 1 - 6 T + 44 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.19.ag_bs
29$D_{4}$ \( 1 - 4 T + 50 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.29.ae_by
31$D_{4}$ \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.31.am_di
37$D_{4}$ \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.37.i_da
41$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.41.a_bi
43$D_{4}$ \( 1 - 2 T + 84 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.43.ac_dg
47$D_{4}$ \( 1 - 8 T + 98 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.47.ai_du
53$D_{4}$ \( 1 - 18 T + 160 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.53.as_ge
59$D_{4}$ \( 1 + 8 T + 122 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.59.i_es
61$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.61.i_be
67$D_{4}$ \( 1 - 2 T + 108 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.67.ac_ee
71$D_{4}$ \( 1 - 8 T + 110 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.71.ai_eg
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$D_{4}$ \( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.79.g_u
83$D_{4}$ \( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_dq
89$D_{4}$ \( 1 + 2 T + 152 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.89.c_fw
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
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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

−8.943631787079980133046764152776, −8.538431317609105813111178456003, −7.949135937159466024533956425667, −7.81070184274993681055779386070, −7.23474088302826110508000128835, −7.02246836257111682043329149661, −6.51065818631839355556928496348, −6.27612768657595080142150099801, −5.76394227373370593510270701998, −5.40589172267718608607241387584, −4.94426350691585473689737632700, −4.73920244061417925386106770548, −4.13734625528361524652657666064, −3.91226433063958838839710346094, −3.00206341757703335567652322263, −2.91398691011172597456758026000, −2.18633763731335165855465608911, −1.79553526367141065555453408579, −1.24150760468322724442998145939, −0.66607370832677637278409282824, 0.66607370832677637278409282824, 1.24150760468322724442998145939, 1.79553526367141065555453408579, 2.18633763731335165855465608911, 2.91398691011172597456758026000, 3.00206341757703335567652322263, 3.91226433063958838839710346094, 4.13734625528361524652657666064, 4.73920244061417925386106770548, 4.94426350691585473689737632700, 5.40589172267718608607241387584, 5.76394227373370593510270701998, 6.27612768657595080142150099801, 6.51065818631839355556928496348, 7.02246836257111682043329149661, 7.23474088302826110508000128835, 7.81070184274993681055779386070, 7.949135937159466024533956425667, 8.538431317609105813111178456003, 8.943631787079980133046764152776

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