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Properties

Label 4-9464e2-1.1-c1e2-0-9
Degree $4$
Conductor $89567296$
Sign $1$
Analytic cond. $5710.88$
Root an. cond. $8.69312$
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  + 3-s + 5·5-s − 2·7-s − 4·9-s − 11-s + 5·15-s − 2·17-s − 19-s − 2·21-s + 4·23-s + 10·25-s − 6·27-s − 5·29-s − 2·31-s − 33-s − 10·35-s − 10·41-s − 3·43-s − 20·45-s − 6·47-s + 3·49-s − 2·51-s + 4·53-s − 5·55-s − 57-s − 10·59-s − 12·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 2.23·5-s − 0.755·7-s − 4/3·9-s − 0.301·11-s + 1.29·15-s − 0.485·17-s − 0.229·19-s − 0.436·21-s + 0.834·23-s + 2·25-s − 1.15·27-s − 0.928·29-s − 0.359·31-s − 0.174·33-s − 1.69·35-s − 1.56·41-s − 0.457·43-s − 2.98·45-s − 0.875·47-s + 3/7·49-s − 0.280·51-s + 0.549·53-s − 0.674·55-s − 0.132·57-s − 1.30·59-s − 1.53·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89567296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89567296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(89567296\)    =    \(2^{6} \cdot 7^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(5710.88\)
Root analytic conductor: \(8.69312\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 89567296,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
13 \( 1 \)
good3$D_{4}$ \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_f
5$C_4$ \( 1 - p T + 3 p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) 2.5.af_p
11$C_4$ \( 1 + T - 9 T^{2} + p T^{3} + p^{2} T^{4} \) 2.11.b_aj
17$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.17.c_bj
19$D_{4}$ \( 1 + T + 27 T^{2} + p T^{3} + p^{2} T^{4} \) 2.19.b_bb
23$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.23.ae_by
29$D_{4}$ \( 1 + 5 T + 33 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.29.f_bh
31$D_{4}$ \( 1 + 2 T + 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.31.c_br
37$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.37.a_ag
41$D_{4}$ \( 1 + 10 T + 102 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.41.k_dy
43$D_{4}$ \( 1 + 3 T + 57 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.43.d_cf
47$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.47.g_dz
53$D_{4}$ \( 1 - 4 T + 65 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.53.ae_cn
59$D_{4}$ \( 1 + 10 T + 123 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.59.k_et
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.61.m_gc
67$D_{4}$ \( 1 - 2 T + 115 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.67.ac_el
71$D_{4}$ \( 1 + 6 T + 146 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.71.g_fq
73$D_{4}$ \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.73.m_dy
79$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \) 2.79.a_da
83$D_{4}$ \( 1 + 2 T + 147 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.83.c_fr
89$D_{4}$ \( 1 - 15 T + 173 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.89.ap_gr
97$D_{4}$ \( 1 - 13 T + 135 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.97.an_ff
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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

−7.58113412484521511100125280469, −7.09067758253493938375293915227, −6.64333142870835793623114525545, −6.52748088050374198071114846120, −6.11888710956022647744871311248, −5.82283827035368104839948473153, −5.44728490804260760132908332222, −5.43176484771495933035614540294, −4.76553430838002997016891632023, −4.60492194738162249879632078574, −3.82804424819168501911569719782, −3.46432497588690018978431996770, −3.03824733364693295702465828944, −2.88713981860330029752294120556, −2.29749775324832457600724185752, −2.13050463280711677456050411415, −1.61112925864846247875673134891, −1.23839142555331160120258544757, 0, 0, 1.23839142555331160120258544757, 1.61112925864846247875673134891, 2.13050463280711677456050411415, 2.29749775324832457600724185752, 2.88713981860330029752294120556, 3.03824733364693295702465828944, 3.46432497588690018978431996770, 3.82804424819168501911569719782, 4.60492194738162249879632078574, 4.76553430838002997016891632023, 5.43176484771495933035614540294, 5.44728490804260760132908332222, 5.82283827035368104839948473153, 6.11888710956022647744871311248, 6.52748088050374198071114846120, 6.64333142870835793623114525545, 7.09067758253493938375293915227, 7.58113412484521511100125280469

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