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Properties

Label 4-2106e2-1.1-c1e2-0-16
Degree $4$
Conductor $4435236$
Sign $1$
Analytic cond. $282.794$
Root an. cond. $4.10079$
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-s − 5-s − 7-s − 8-s − 10-s − 2·11-s + 13-s − 14-s − 16-s + 6·17-s + 12·19-s − 2·22-s − 4·23-s + 5·25-s + 26-s + 2·29-s − 4·31-s + 6·34-s + 35-s + 6·37-s + 12·38-s + 40-s + 5·43-s − 4·46-s + 13·47-s + 7·49-s + 5·50-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s − 0.603·11-s + 0.277·13-s − 0.267·14-s − 1/4·16-s + 1.45·17-s + 2.75·19-s − 0.426·22-s − 0.834·23-s + 25-s + 0.196·26-s + 0.371·29-s − 0.718·31-s + 1.02·34-s + 0.169·35-s + 0.986·37-s + 1.94·38-s + 0.158·40-s + 0.762·43-s − 0.589·46-s + 1.89·47-s + 49-s + 0.707·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4435236 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4435236 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4435236\)    =    \(2^{2} \cdot 3^{8} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(282.794\)
Root analytic conductor: \(4.10079\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4435236,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.254622356\)
\(L(\frac12)\) \(\approx\) \(3.254622356\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3 \( 1 \)
13$C_2$ \( 1 - T + T^{2} \)
good5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_ae
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.7.b_ag
11$C_2^2$ \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_ah
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.17.ag_br
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.19.am_cw
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.23.e_ah
29$C_2^2$ \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.29.ac_az
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.31.e_ap
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.37.ag_df
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.41.a_abp
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.af_as
47$C_2^2$ \( 1 - 13 T + 122 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.47.an_es
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.53.y_jq
59$C_2^2$ \( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.59.k_bp
61$C_2^2$ \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.61.ai_d
67$C_2^2$ \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.67.ac_acl
71$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.71.ak_gl
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.73.u_jm
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.79.ae_acl
83$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.83.a_adf
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) 2.97.o_dv
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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

−9.278618396668619124811502331646, −9.056011054004284689535776837013, −8.346478494285895980672445006642, −8.155158357871917563937295842750, −7.48720107403833107940490488327, −7.42423383851981146533074821641, −7.22315518475479264486990521188, −6.36929858883311238581921757481, −6.01606127851934128848665822470, −5.53277647712484905292835912982, −5.52834017316607532442378663378, −4.78826031533615262557336386055, −4.54199999114513792529513647134, −3.93482226144401732293103750198, −3.42193915554129934580704945431, −2.99913687686733076203081390454, −2.97230593924549699693463898791, −1.99954717442789171392254787456, −1.14540419468181885280899774265, −0.66423205213191386813545210556, 0.66423205213191386813545210556, 1.14540419468181885280899774265, 1.99954717442789171392254787456, 2.97230593924549699693463898791, 2.99913687686733076203081390454, 3.42193915554129934580704945431, 3.93482226144401732293103750198, 4.54199999114513792529513647134, 4.78826031533615262557336386055, 5.52834017316607532442378663378, 5.53277647712484905292835912982, 6.01606127851934128848665822470, 6.36929858883311238581921757481, 7.22315518475479264486990521188, 7.42423383851981146533074821641, 7.48720107403833107940490488327, 8.155158357871917563937295842750, 8.346478494285895980672445006642, 9.056011054004284689535776837013, 9.278618396668619124811502331646

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