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Properties

Label 4-756e2-1.1-c1e2-0-8
Degree $4$
Conductor $571536$
Sign $1$
Analytic cond. $36.4416$
Root an. cond. $2.45696$
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  − 5·7-s + 15·19-s + 5·25-s + 15·31-s − 10·37-s − 10·43-s + 18·49-s − 27·61-s + 16·67-s + 3·73-s + 4·79-s − 6·103-s + 17·109-s − 11·121-s + 127-s + 131-s − 75·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + 173-s − 25·175-s + ⋯
L(s)  = 1  − 1.88·7-s + 3.44·19-s + 25-s + 2.69·31-s − 1.64·37-s − 1.52·43-s + 18/7·49-s − 3.45·61-s + 1.95·67-s + 0.351·73-s + 0.450·79-s − 0.591·103-s + 1.62·109-s − 121-s + 0.0887·127-s + 0.0873·131-s − 6.50·133-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 + 0.0760·173-s − 1.88·175-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(571536\)    =    \(2^{4} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(36.4416\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 571536,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.551076189\)
\(L(\frac12)\) \(\approx\) \(1.551076189\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + 5 T + p T^{2} \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.5.a_af
11$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.11.a_l
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) 2.19.ap_dq
23$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.23.a_x
29$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.29.a_acg
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.31.ap_ec
37$C_2$ \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.37.k_cl
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.43.k_eh
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.53.a_cb
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2$ \( ( 1 + 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.bb_ls
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.67.aq_hh
71$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.71.a_afm
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.73.ad_cy
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.79.ae_acl
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.89.a_adl
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) 2.97.a_agl
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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

−10.27433520192045755536409980540, −10.17982268855402945013562671045, −9.652152822443171192462220552798, −9.417328076287409178368212643355, −8.998087384106584395900630749474, −8.513173714042184614171648711457, −7.82733515212703333505558598603, −7.59478046444674834901872276718, −6.90413400507529591911480306607, −6.69048801424228202748010266557, −6.28760857130199448742308190491, −5.69511910136387650918038762607, −5.07156293582988964815464365744, −4.93785102777930471911825444536, −3.97161236065414384806823118395, −3.35892479444412213172694053682, −3.03776093403304463992609751991, −2.76143900328984387950086327504, −1.45093610757812644631477506142, −0.68566402181632143400936868225, 0.68566402181632143400936868225, 1.45093610757812644631477506142, 2.76143900328984387950086327504, 3.03776093403304463992609751991, 3.35892479444412213172694053682, 3.97161236065414384806823118395, 4.93785102777930471911825444536, 5.07156293582988964815464365744, 5.69511910136387650918038762607, 6.28760857130199448742308190491, 6.69048801424228202748010266557, 6.90413400507529591911480306607, 7.59478046444674834901872276718, 7.82733515212703333505558598603, 8.513173714042184614171648711457, 8.998087384106584395900630749474, 9.417328076287409178368212643355, 9.652152822443171192462220552798, 10.17982268855402945013562671045, 10.27433520192045755536409980540

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