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Properties

Label 4-8624e2-1.1-c1e2-0-14
Degree $4$
Conductor $74373376$
Sign $1$
Analytic cond. $4742.11$
Root an. cond. $8.29837$
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  + 3-s − 3·5-s − 9-s + 2·11-s + 2·13-s − 3·15-s − 4·17-s − 8·19-s − 9·23-s + 25-s − 2·29-s − 7·31-s + 2·33-s − 11·37-s + 2·39-s − 6·41-s + 6·43-s + 3·45-s + 16·47-s − 4·51-s + 8·53-s − 6·55-s − 8·57-s − 5·59-s + 6·61-s − 6·65-s − 15·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.34·5-s − 1/3·9-s + 0.603·11-s + 0.554·13-s − 0.774·15-s − 0.970·17-s − 1.83·19-s − 1.87·23-s + 1/5·25-s − 0.371·29-s − 1.25·31-s + 0.348·33-s − 1.80·37-s + 0.320·39-s − 0.937·41-s + 0.914·43-s + 0.447·45-s + 2.33·47-s − 0.560·51-s + 1.09·53-s − 0.809·55-s − 1.05·57-s − 0.650·59-s + 0.768·61-s − 0.744·65-s − 1.83·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74373376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74373376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(74373376\)    =    \(2^{8} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(4742.11\)
Root analytic conductor: \(8.29837\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 74373376,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
7 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_c
5$C_2^2$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.5.d_i
13$C_4$ \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.13.ac_k
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.17.e_bm
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$D_{4}$ \( 1 + 9 T + 62 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.23.j_ck
29$D_{4}$ \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.29.c_bq
31$D_{4}$ \( 1 + 7 T + 70 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.31.h_cs
37$D_{4}$ \( 1 + 11 T + 100 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.37.l_dw
41$D_{4}$ \( 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_cw
43$D_{4}$ \( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.43.ag_da
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.47.aq_gc
53$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.53.ai_cc
59$D_{4}$ \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.59.f_s
61$D_{4}$ \( 1 - 6 T + 114 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.61.ag_ek
67$D_{4}$ \( 1 + 15 T + 186 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.67.p_he
71$D_{4}$ \( 1 - 5 T + 110 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.71.af_eg
73$D_{4}$ \( 1 + 2 T + 130 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_fa
79$D_{4}$ \( 1 - 14 T + 190 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.79.ao_hi
83$D_{4}$ \( 1 - 10 T + 174 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.83.ak_gs
89$D_{4}$ \( 1 - 7 T + 152 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.89.ah_fw
97$D_{4}$ \( 1 + 27 T + 372 T^{2} + 27 p T^{3} + p^{2} T^{4} \) 2.97.bb_oi
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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

−7.72645573277955464689090199691, −7.29985050209686827612658759846, −6.81670253521833227305365473381, −6.81359986000869526084346729084, −6.22308455262560822816572349667, −5.96563997839194957250398797818, −5.51158118552307337121038958690, −5.28210978315635070318475252787, −4.44993997262429244241923068084, −4.34469527156392098668235461347, −3.93602802262462089723564526126, −3.85047584102653744113551319823, −3.41041385135408346467192367714, −2.97105075492248133696782673363, −2.19378286884693702259011406395, −2.14224078262278293648941240445, −1.73108148196738799400638989997, −0.866503153579054288206624202194, 0, 0, 0.866503153579054288206624202194, 1.73108148196738799400638989997, 2.14224078262278293648941240445, 2.19378286884693702259011406395, 2.97105075492248133696782673363, 3.41041385135408346467192367714, 3.85047584102653744113551319823, 3.93602802262462089723564526126, 4.34469527156392098668235461347, 4.44993997262429244241923068084, 5.28210978315635070318475252787, 5.51158118552307337121038958690, 5.96563997839194957250398797818, 6.22308455262560822816572349667, 6.81359986000869526084346729084, 6.81670253521833227305365473381, 7.29985050209686827612658759846, 7.72645573277955464689090199691

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