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Properties

Label 4-2952e2-1.1-c1e2-0-1
Degree $4$
Conductor $8714304$
Sign $1$
Analytic cond. $555.631$
Root an. cond. $4.85508$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s − 4·7-s − 2·11-s − 4·13-s + 2·17-s − 8·19-s + 4·25-s + 10·29-s − 10·31-s − 16·35-s − 10·37-s − 2·41-s − 14·43-s + 6·47-s − 8·53-s − 8·55-s − 8·59-s − 6·61-s − 16·65-s − 4·67-s + 2·71-s + 18·73-s + 8·77-s − 12·79-s − 12·83-s + 8·85-s − 12·89-s + ⋯
L(s)  = 1  + 1.78·5-s − 1.51·7-s − 0.603·11-s − 1.10·13-s + 0.485·17-s − 1.83·19-s + 4/5·25-s + 1.85·29-s − 1.79·31-s − 2.70·35-s − 1.64·37-s − 0.312·41-s − 2.13·43-s + 0.875·47-s − 1.09·53-s − 1.07·55-s − 1.04·59-s − 0.768·61-s − 1.98·65-s − 0.488·67-s + 0.237·71-s + 2.10·73-s + 0.911·77-s − 1.35·79-s − 1.31·83-s + 0.867·85-s − 1.27·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8714304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8714304 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8714304\)    =    \(2^{6} \cdot 3^{4} \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(555.631\)
Root analytic conductor: \(4.85508\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 8714304,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3 \( 1 \)
41$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.5.ae_m
7$D_{4}$ \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.7.e_q
11$D_{4}$ \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_f
13$D_{4}$ \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.13.e_bc
17$D_{4}$ \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.17.ac_r
19$D_{4}$ \( 1 + 8 T + 36 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.19.i_bk
23$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \) 2.23.a_bc
29$D_{4}$ \( 1 - 10 T + 81 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.29.ak_dd
31$D_{4}$ \( 1 + 10 T + 79 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.31.k_db
37$D_{4}$ \( 1 + 10 T + 91 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.37.k_dn
43$D_{4}$ \( 1 + 14 T + 127 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.43.o_ex
47$D_{4}$ \( 1 - 6 T + 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.47.ag_cb
53$D_{4}$ \( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.53.i_by
59$D_{4}$ \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.59.i_ew
61$D_{4}$ \( 1 + 6 T + 3 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.61.g_d
67$D_{4}$ \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_co
71$D_{4}$ \( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.71.ac_at
73$D_{4}$ \( 1 - 18 T + 195 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.73.as_hn
79$D_{4}$ \( 1 + 12 T + 66 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.79.m_co
83$D_{4}$ \( 1 + 12 T + 200 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_hs
89$D_{4}$ \( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_ha
97$D_{4}$ \( 1 - 16 T + 208 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.97.aq_ia
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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

−8.676880740960376613387435881411, −8.293240725431010817262677317562, −7.74779037545582673472171134512, −7.41473860116791575585546531895, −6.76360634897482216581752673403, −6.66872352777953560319885651172, −6.24820719370781012601687990985, −6.07589736571034572443159539980, −5.38682397195424098083717111414, −5.27237043892368107726940301251, −4.78121152981359308754003916176, −4.33517640250829996137560961106, −3.53327523616376937094568011108, −3.34710536512179828565725679470, −2.69717028535886628952237833429, −2.41577627893973314069773073338, −1.80797734262899772879786149026, −1.50425541198669171516645447646, 0, 0, 1.50425541198669171516645447646, 1.80797734262899772879786149026, 2.41577627893973314069773073338, 2.69717028535886628952237833429, 3.34710536512179828565725679470, 3.53327523616376937094568011108, 4.33517640250829996137560961106, 4.78121152981359308754003916176, 5.27237043892368107726940301251, 5.38682397195424098083717111414, 6.07589736571034572443159539980, 6.24820719370781012601687990985, 6.66872352777953560319885651172, 6.76360634897482216581752673403, 7.41473860116791575585546531895, 7.74779037545582673472171134512, 8.293240725431010817262677317562, 8.676880740960376613387435881411

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