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Properties

Label 4-9522e2-1.1-c1e2-0-8
Degree $4$
Conductor $90668484$
Sign $1$
Analytic cond. $5781.10$
Root an. cond. $8.71972$
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  + 2·2-s + 3·4-s + 4·8-s + 12·13-s + 5·16-s − 2·25-s + 24·26-s − 4·29-s + 16·31-s + 6·32-s + 12·41-s − 16·47-s − 6·49-s − 4·50-s + 36·52-s − 8·58-s − 8·59-s + 32·62-s + 7·64-s − 16·71-s + 12·73-s + 24·82-s − 32·94-s − 12·98-s − 6·100-s + 20·101-s + 48·104-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.41·8-s + 3.32·13-s + 5/4·16-s − 2/5·25-s + 4.70·26-s − 0.742·29-s + 2.87·31-s + 1.06·32-s + 1.87·41-s − 2.33·47-s − 6/7·49-s − 0.565·50-s + 4.99·52-s − 1.05·58-s − 1.04·59-s + 4.06·62-s + 7/8·64-s − 1.89·71-s + 1.40·73-s + 2.65·82-s − 3.30·94-s − 1.21·98-s − 3/5·100-s + 1.99·101-s + 4.70·104-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90668484 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90668484 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(90668484\)    =    \(2^{2} \cdot 3^{4} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(5781.10\)
Root analytic conductor: \(8.71972\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 90668484,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(13.69780911\)
\(L(\frac12)\) \(\approx\) \(13.69780911\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
23 \( 1 \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.7.a_g
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.13.am_ck
17$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.17.a_ba
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.19.a_abi
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.31.aq_ew
37$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.37.a_cw
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \) 2.43.a_da
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.47.q_gc
53$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.53.a_bi
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.61.a_es
67$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.67.a_ck
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.71.q_hy
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2^2$ \( 1 + 150 T^{2} + p^{2} T^{4} \) 2.79.a_fu
83$C_2^2$ \( 1 + 134 T^{2} + p^{2} T^{4} \) 2.83.a_fe
89$C_2^2$ \( 1 + 170 T^{2} + p^{2} T^{4} \) 2.89.a_go
97$C_2^2$ \( 1 + 162 T^{2} + p^{2} T^{4} \) 2.97.a_gg
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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

−7.78921552137529473568248656094, −7.64658135438301618809106208451, −6.87089234113473461019458434546, −6.65946541497931752144823284446, −6.38453136038632894148927603590, −6.04334901080074132728976002763, −5.93872193253636019596015585552, −5.59257423497698686146590578151, −5.06811053624052941433687430277, −4.57097329752742631798831532279, −4.42342175992848786610749414020, −4.07565733374831554159795470969, −3.60335718785474470521827675362, −3.30848599816694871509850352216, −3.05094368964784861351594620847, −2.64685363103871702704511844801, −1.82325876393217757814427777617, −1.71835201091011355915175368080, −1.08584830647055053610167509561, −0.69817369664604250788404374927, 0.69817369664604250788404374927, 1.08584830647055053610167509561, 1.71835201091011355915175368080, 1.82325876393217757814427777617, 2.64685363103871702704511844801, 3.05094368964784861351594620847, 3.30848599816694871509850352216, 3.60335718785474470521827675362, 4.07565733374831554159795470969, 4.42342175992848786610749414020, 4.57097329752742631798831532279, 5.06811053624052941433687430277, 5.59257423497698686146590578151, 5.93872193253636019596015585552, 6.04334901080074132728976002763, 6.38453136038632894148927603590, 6.65946541497931752144823284446, 6.87089234113473461019458434546, 7.64658135438301618809106208451, 7.78921552137529473568248656094

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