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Properties

Label 4-88e2-1.1-c1e2-0-1
Degree $4$
Conductor $7744$
Sign $1$
Analytic cond. $0.493764$
Root an. cond. $0.838262$
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  + 3-s + 3·5-s − 2·7-s − 9-s − 2·11-s − 2·13-s + 3·15-s + 4·17-s − 8·19-s − 2·21-s + 9·23-s + 25-s − 2·29-s − 7·31-s − 2·33-s − 6·35-s − 11·37-s − 2·39-s + 6·41-s − 6·43-s − 3·45-s + 16·47-s + 6·49-s + 4·51-s + 8·53-s − 6·55-s − 8·57-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.34·5-s − 0.755·7-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 − 0.436·21-s + 1.87·23-s + 1/5·25-s − 0.371·29-s − 1.25·31-s − 0.348·33-s − 1.01·35-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 + 6/7·49-s + 0.560·51-s + 1.09·53-s − 0.809·55-s − 1.05·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7744\)    =    \(2^{6} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(0.493764\)
Root analytic conductor: \(0.838262\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7744,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.161238533\)
\(L(\frac12)\) \(\approx\) \(1.161238533\)
\(L(\frac{3}{2})\) 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 \( 1 \)
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.ad_i
7$D_{4}$ \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_ac
13$C_4$ \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.13.c_k
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.17.ae_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.aj_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.ag_cw
43$D_{4}$ \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.43.g_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.g_ek
67$D_{4}$ \( 1 - 15 T + 186 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.67.ap_he
71$D_{4}$ \( 1 + 5 T + 110 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.71.f_eg
73$D_{4}$ \( 1 - 2 T + 130 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.73.ac_fa
79$D_{4}$ \( 1 + 14 T + 190 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.79.o_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.h_fw
97$D_{4}$ \( 1 - 27 T + 372 T^{2} - 27 p T^{3} + p^{2} T^{4} \) 2.97.abb_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

−14.28030020696909154085974651898, −14.03028365827802226978533457695, −13.21360250199287938415795898000, −13.07157866190739138157139780231, −12.50420640112352694735407160234, −11.98337737607109060673329523815, −10.85862827889564040963379281665, −10.64132391117952879925337790245, −10.03086860437715033812573043024, −9.493335556861131114975513791281, −8.764378598884747806441631931744, −8.718623945860768571960683269952, −7.48612764691425437549580882573, −7.09147745434457443207327568161, −6.20506715360269760896144970392, −5.62373588893482846993949544263, −5.05638461162604557379224075722, −3.80500596388585644052637569942, −2.85432814896664270886895797804, −2.11123763103283648443805785938, 2.11123763103283648443805785938, 2.85432814896664270886895797804, 3.80500596388585644052637569942, 5.05638461162604557379224075722, 5.62373588893482846993949544263, 6.20506715360269760896144970392, 7.09147745434457443207327568161, 7.48612764691425437549580882573, 8.718623945860768571960683269952, 8.764378598884747806441631931744, 9.493335556861131114975513791281, 10.03086860437715033812573043024, 10.64132391117952879925337790245, 10.85862827889564040963379281665, 11.98337737607109060673329523815, 12.50420640112352694735407160234, 13.07157866190739138157139780231, 13.21360250199287938415795898000, 14.03028365827802226978533457695, 14.28030020696909154085974651898

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