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Properties

Label 4-34e4-1.1-c1e2-0-0
Degree $4$
Conductor $1336336$
Sign $1$
Analytic cond. $85.2059$
Root an. cond. $3.03820$
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  − 4·9-s − 8·13-s + 8·19-s − 2·25-s − 16·43-s − 24·47-s + 4·49-s + 12·53-s − 8·67-s + 7·81-s + 24·89-s + 24·101-s + 16·103-s + 32·117-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s − 32·171-s + 173-s + ⋯
L(s)  = 1  − 4/3·9-s − 2.21·13-s + 1.83·19-s − 2/5·25-s − 2.43·43-s − 3.50·47-s + 4/7·49-s + 1.64·53-s − 0.977·67-s + 7/9·81-s + 2.54·89-s + 2.38·101-s + 1.57·103-s + 2.95·117-s − 1.81·121-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 − 2.44·171-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1336336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1336336\)    =    \(2^{4} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(85.2059\)
Root analytic conductor: \(3.03820\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1336336,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8038169736\)
\(L(\frac12)\) \(\approx\) \(0.8038169736\)
\(L(\frac{3}{2})\) 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 \( 1 \)
17 \( 1 \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.7.a_ae
11$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.11.a_u
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.13.i_bq
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \) 2.23.a_bs
29$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.29.a_by
31$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \) 2.31.a_bs
37$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.37.a_c
41$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.41.a_abu
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.43.q_fu
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.47.y_je
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.61.a_by
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2^2$ \( 1 + 92 T^{2} + p^{2} T^{4} \) 2.71.a_do
73$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.73.a_fq
79$C_2^2$ \( 1 + 140 T^{2} + p^{2} T^{4} \) 2.79.a_fk
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.89.ay_mk
97$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.97.a_hm
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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

−9.820877248678741021078435099738, −9.806659782136573992683484014266, −9.153738336732082472298771937418, −8.850512460665911147027130168836, −8.338225501505206698232318866957, −7.901294660531445487099182815004, −7.46344545170066672053081768118, −7.33946890035512570644735160509, −6.49734584134990517404672004472, −6.42271118336186390398577121267, −5.61934534289812197648321761758, −5.22872066783070798966580779269, −4.94950968454369140916999574618, −4.64936564913410898075509567553, −3.54592161603304460229851009899, −3.38064501830856532421279783834, −2.76411928992495034613453385642, −2.28118485285684843309267136820, −1.55847171028009317779759272436, −0.37814415421150650269067730868, 0.37814415421150650269067730868, 1.55847171028009317779759272436, 2.28118485285684843309267136820, 2.76411928992495034613453385642, 3.38064501830856532421279783834, 3.54592161603304460229851009899, 4.64936564913410898075509567553, 4.94950968454369140916999574618, 5.22872066783070798966580779269, 5.61934534289812197648321761758, 6.42271118336186390398577121267, 6.49734584134990517404672004472, 7.33946890035512570644735160509, 7.46344545170066672053081768118, 7.901294660531445487099182815004, 8.338225501505206698232318866957, 8.850512460665911147027130168836, 9.153738336732082472298771937418, 9.806659782136573992683484014266, 9.820877248678741021078435099738

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