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Properties

Label 4-2106e2-1.1-c1e2-0-23
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 + 2·5-s − 7-s + 8-s − 2·10-s + 5·11-s − 13-s + 14-s − 16-s − 4·17-s + 2·19-s − 5·22-s + 5·25-s + 26-s + 9·29-s + 4·31-s + 4·34-s − 2·35-s + 8·37-s − 2·38-s + 2·40-s − 4·41-s + 2·43-s + 8·47-s + 7·49-s − 5·50-s + 2·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.894·5-s − 0.377·7-s + 0.353·8-s − 0.632·10-s + 1.50·11-s − 0.277·13-s + 0.267·14-s − 1/4·16-s − 0.970·17-s + 0.458·19-s − 1.06·22-s + 25-s + 0.196·26-s + 1.67·29-s + 0.718·31-s + 0.685·34-s − 0.338·35-s + 1.31·37-s − 0.324·38-s + 0.316·40-s − 0.624·41-s + 0.304·43-s + 1.16·47-s + 49-s − 0.707·50-s + 0.274·53-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

\(L(1)\) \(\approx\) \(2.324595502\)
\(L(\frac12)\) \(\approx\) \(2.324595502\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.5.ac_ab
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.7.b_ag
11$C_2^2$ \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.11.af_o
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.17.e_bm
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.19.ac_bn
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.23.a_ax
29$C_2^2$ \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.29.aj_ca
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.ae_ap
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.37.ai_dm
41$C_2^2$ \( 1 + 4 T - 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.41.e_az
43$C_2^2$ \( 1 - 2 T - 39 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.43.ac_abn
47$C_2^2$ \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.47.ai_r
53$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.53.ac_ed
59$C_2^2$ \( 1 + 5 T - 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.59.f_abi
61$C_2^2$ \( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.61.al_ci
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_abz
71$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.71.ba_lz
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.73.i_gg
79$C_2^2$ \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.79.o_en
83$C_2^2$ \( 1 - 3 T - 74 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.83.ad_acw
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.89.e_ha
97$C_2^2$ \( 1 - 8 T - 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.97.ai_abh
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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

−9.091488398697977682708868599428, −8.943197550799740161073163405564, −8.655194180678767347828646410400, −8.502157200079031353622829936754, −7.55696725016545011668446828565, −7.46765937070494437290041916474, −6.91727786893683457739090827042, −6.61147952187297266094383416091, −6.20748364860750203997990702114, −5.87267124577512591624655370291, −5.48722853631335022386175763306, −4.68105475857761440023683994644, −4.33926786257199685290699030768, −4.31888656906155238767280557902, −3.23243574260454292145072011109, −3.06431372297802439831580570033, −2.30206087023700909515427407763, −1.91292813620218371756274348274, −1.06922782308171936659645161051, −0.74628006500612920951448869724, 0.74628006500612920951448869724, 1.06922782308171936659645161051, 1.91292813620218371756274348274, 2.30206087023700909515427407763, 3.06431372297802439831580570033, 3.23243574260454292145072011109, 4.31888656906155238767280557902, 4.33926786257199685290699030768, 4.68105475857761440023683994644, 5.48722853631335022386175763306, 5.87267124577512591624655370291, 6.20748364860750203997990702114, 6.61147952187297266094383416091, 6.91727786893683457739090827042, 7.46765937070494437290041916474, 7.55696725016545011668446828565, 8.502157200079031353622829936754, 8.655194180678767347828646410400, 8.943197550799740161073163405564, 9.091488398697977682708868599428

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