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Properties

Label 4-364e2-1.1-c1e2-0-11
Degree $4$
Conductor $132496$
Sign $1$
Analytic cond. $8.44805$
Root an. cond. $1.70486$
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  + 6·5-s − 7-s + 3·9-s + 2·11-s − 5·13-s + 7·17-s − 2·19-s + 4·23-s + 17·25-s − 29-s + 8·31-s − 6·35-s − 37-s + 3·41-s + 6·43-s + 18·45-s − 20·47-s − 14·53-s + 12·55-s − 6·59-s − 7·61-s − 3·63-s − 30·65-s − 8·67-s + 6·71-s − 22·73-s − 2·77-s + ⋯
L(s)  = 1  + 2.68·5-s − 0.377·7-s + 9-s + 0.603·11-s − 1.38·13-s + 1.69·17-s − 0.458·19-s + 0.834·23-s + 17/5·25-s − 0.185·29-s + 1.43·31-s − 1.01·35-s − 0.164·37-s + 0.468·41-s + 0.914·43-s + 2.68·45-s − 2.91·47-s − 1.92·53-s + 1.61·55-s − 0.781·59-s − 0.896·61-s − 0.377·63-s − 3.72·65-s − 0.977·67-s + 0.712·71-s − 2.57·73-s − 0.227·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(132496\)    =    \(2^{4} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(8.44805\)
Root analytic conductor: \(1.70486\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 132496,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.802267355\)
\(L(\frac12)\) \(\approx\) \(2.802267355\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + T + T^{2} \)
13$C_2$ \( 1 + 5 T + p T^{2} \)
good3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) 2.3.a_ad
5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.5.ag_t
11$C_2^2$ \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_ah
17$C_2^2$ \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.17.ah_bg
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_ap
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_ah
29$C_2^2$ \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \) 2.29.b_abc
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.37.b_abk
41$C_2^2$ \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.41.ad_abg
43$C_2^2$ \( 1 - 6 T - 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.43.ag_ah
47$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.47.u_hm
53$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.53.o_fz
59$C_2^2$ \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.59.g_ax
61$C_2^2$ \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.61.h_am
67$C_2^2$ \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_ad
71$C_2^2$ \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.71.ag_abj
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.73.w_kh
79$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.79.bc_nq
83$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.83.bc_ny
89$C_2^2$ \( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.89.ak_l
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.97.c_adp
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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

−11.64492013222373305915677248092, −11.18520933320882636094504571450, −10.23431374179834033598698996549, −10.11657093232847075288651127355, −9.937329806638096280548667497977, −9.593727841148534079709917470429, −9.101845365791697172004139227886, −8.654453126504639617468003886054, −7.69198551631788915066343785741, −7.44551575061954126480534635047, −6.60295912425889250865853283146, −6.47924636163764428642298671077, −5.66354771492519068520748451504, −5.65707523803847805723905882179, −4.57413443765668782587013113290, −4.56519468779227012781153479697, −2.96449856701690682339762277180, −2.95105732575103744411499322197, −1.60022620292165731878648412676, −1.57827748364288420912625091633, 1.57827748364288420912625091633, 1.60022620292165731878648412676, 2.95105732575103744411499322197, 2.96449856701690682339762277180, 4.56519468779227012781153479697, 4.57413443765668782587013113290, 5.65707523803847805723905882179, 5.66354771492519068520748451504, 6.47924636163764428642298671077, 6.60295912425889250865853283146, 7.44551575061954126480534635047, 7.69198551631788915066343785741, 8.654453126504639617468003886054, 9.101845365791697172004139227886, 9.593727841148534079709917470429, 9.937329806638096280548667497977, 10.11657093232847075288651127355, 10.23431374179834033598698996549, 11.18520933320882636094504571450, 11.64492013222373305915677248092

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