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Properties

Label 4-5712e2-1.1-c1e2-0-9
Degree $4$
Conductor $32626944$
Sign $1$
Analytic cond. $2080.32$
Root an. cond. $6.75355$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 3·5-s − 2·7-s + 3·9-s − 3·11-s + 13-s + 6·15-s + 2·17-s − 4·19-s + 4·21-s + 6·23-s + 5·25-s − 4·27-s + 6·29-s − 16·31-s + 6·33-s + 6·35-s + 7·37-s − 2·39-s − 6·41-s − 7·43-s − 9·45-s + 3·49-s − 4·51-s + 3·53-s + 9·55-s + 8·57-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.34·5-s − 0.755·7-s + 9-s − 0.904·11-s + 0.277·13-s + 1.54·15-s + 0.485·17-s − 0.917·19-s + 0.872·21-s + 1.25·23-s + 25-s − 0.769·27-s + 1.11·29-s − 2.87·31-s + 1.04·33-s + 1.01·35-s + 1.15·37-s − 0.320·39-s − 0.937·41-s − 1.06·43-s − 1.34·45-s + 3/7·49-s − 0.560·51-s + 0.412·53-s + 1.21·55-s + 1.05·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32626944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32626944 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(32626944\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(2080.32\)
Root analytic conductor: \(6.75355\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 32626944,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
17$C_1$ \( ( 1 - T )^{2} \)
good5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.5.d_e
11$D_{4}$ \( 1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_q
13$D_{4}$ \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) 2.13.ab_s
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.19.e_bq
23$D_{4}$ \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_w
29$D_{4}$ \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.29.ag_bi
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.31.q_ew
37$D_{4}$ \( 1 - 7 T + 78 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.37.ah_da
41$D_{4}$ \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_cg
43$D_{4}$ \( 1 + 7 T + 90 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.43.h_dm
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$D_{4}$ \( 1 - 3 T + 100 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.53.ad_dw
59$D_{4}$ \( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.59.g_dq
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$D_{4}$ \( 1 + T + 126 T^{2} + p T^{3} + p^{2} T^{4} \) 2.67.b_ew
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$D_{4}$ \( 1 - 7 T + 84 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.73.ah_dg
79$D_{4}$ \( 1 + 7 T + 162 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.79.h_gg
83$D_{4}$ \( 1 - 15 T + 148 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.83.ap_fs
89$D_{4}$ \( 1 - 9 T + 124 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.89.aj_eu
97$C_2^2$ \( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} \) 2.97.r_hk
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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

−7.73361887612881136652914033128, −7.64221647061547974806344745227, −7.18530602752001091787139755510, −6.86113750709348235866530374526, −6.56811559952778590700930687023, −6.24672266095772684540076642313, −5.75816092935465498805275204569, −5.37372507105355811644174497669, −5.02762846400117920809429997438, −4.84894709022249698169384630205, −4.18423711371031854527468907151, −3.96998930405618469738071453089, −3.44537484219913477403185169099, −3.24652447783238928017247507670, −2.63938156536672555404900057527, −2.13189504087356797400154286994, −1.38651459628156348440289711404, −0.861540058274778257144946014590, 0, 0, 0.861540058274778257144946014590, 1.38651459628156348440289711404, 2.13189504087356797400154286994, 2.63938156536672555404900057527, 3.24652447783238928017247507670, 3.44537484219913477403185169099, 3.96998930405618469738071453089, 4.18423711371031854527468907151, 4.84894709022249698169384630205, 5.02762846400117920809429997438, 5.37372507105355811644174497669, 5.75816092935465498805275204569, 6.24672266095772684540076642313, 6.56811559952778590700930687023, 6.86113750709348235866530374526, 7.18530602752001091787139755510, 7.64221647061547974806344745227, 7.73361887612881136652914033128

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