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Properties

Label 4-4232e2-1.1-c1e2-0-5
Degree $4$
Conductor $17909824$
Sign $1$
Analytic cond. $1141.94$
Root an. cond. $5.81314$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s − 2·5-s + 8·9-s − 4·11-s + 2·13-s + 8·15-s + 8·17-s + 8·19-s − 7·25-s − 12·27-s + 6·29-s − 12·31-s + 16·33-s − 8·39-s − 2·41-s − 4·43-s − 16·45-s − 4·47-s + 4·49-s − 32·51-s + 10·53-s + 8·55-s − 32·57-s − 8·59-s + 14·61-s − 4·65-s − 8·67-s + ⋯
L(s)  = 1  − 2.30·3-s − 0.894·5-s + 8/3·9-s − 1.20·11-s + 0.554·13-s + 2.06·15-s + 1.94·17-s + 1.83·19-s − 7/5·25-s − 2.30·27-s + 1.11·29-s − 2.15·31-s + 2.78·33-s − 1.28·39-s − 0.312·41-s − 0.609·43-s − 2.38·45-s − 0.583·47-s + 4/7·49-s − 4.48·51-s + 1.37·53-s + 1.07·55-s − 4.23·57-s − 1.04·59-s + 1.79·61-s − 0.496·65-s − 0.977·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17909824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17909824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(17909824\)    =    \(2^{6} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(1141.94\)
Root analytic conductor: \(5.81314\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 17909824,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
23 \( 1 \)
good3$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.3.e_i
5$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.5.c_l
7$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.7.a_ae
11$D_{4}$ \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_y
13$D_{4}$ \( 1 - 2 T + 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.13.ac_t
17$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.17.ai_bq
19$D_{4}$ \( 1 - 8 T + 52 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.19.ai_ca
29$D_{4}$ \( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.29.ag_ch
31$D_{4}$ \( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.31.m_dm
37$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \) 2.37.a_bq
41$D_{4}$ \( 1 + 2 T + 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.41.c_cx
43$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_s
47$D_{4}$ \( 1 + 4 T + 80 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.47.e_dc
53$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.53.ak_fb
59$D_{4}$ \( 1 + 8 T + 84 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.59.i_dg
61$D_{4}$ \( 1 - 14 T + 163 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.61.ao_gh
67$D_{4}$ \( 1 + 8 T + 118 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_eo
71$D_{4}$ \( 1 - 8 T + 108 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.71.ai_ee
73$D_{4}$ \( 1 + 2 T + 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_cx
79$D_{4}$ \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.79.i_bu
83$D_{4}$ \( 1 - 12 T + 194 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_hm
89$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.89.o_it
97$D_{4}$ \( 1 + 14 T + 171 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.97.o_gp
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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

−7.87669600275718346298551928544, −7.77710505130983978037427222543, −7.39780715956686010196059825685, −7.33372540589209196528732218477, −6.52582327214013996161603825823, −6.42222808219751697976773385774, −5.74949874238289106673965402018, −5.50551773325741218719591181767, −5.42671876358889434575316725103, −5.12035922100571593867202195668, −4.68326717101496282479031261263, −3.93823899031924834765176865645, −3.71905882575682884590083760968, −3.39848405792676489834064964485, −2.78219948919496933445156708870, −2.09503563778732560657747316865, −1.14796285619962485411544033116, −1.12595790299866945065869493777, 0, 0, 1.12595790299866945065869493777, 1.14796285619962485411544033116, 2.09503563778732560657747316865, 2.78219948919496933445156708870, 3.39848405792676489834064964485, 3.71905882575682884590083760968, 3.93823899031924834765176865645, 4.68326717101496282479031261263, 5.12035922100571593867202195668, 5.42671876358889434575316725103, 5.50551773325741218719591181767, 5.74949874238289106673965402018, 6.42222808219751697976773385774, 6.52582327214013996161603825823, 7.33372540589209196528732218477, 7.39780715956686010196059825685, 7.77710505130983978037427222543, 7.87669600275718346298551928544

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