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Properties

Label 4-2496e2-1.1-c1e2-0-7
Degree $4$
Conductor $6230016$
Sign $1$
Analytic cond. $397.231$
Root an. cond. $4.46437$
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  + 2·3-s + 3·9-s − 6·13-s − 4·17-s + 16·23-s + 6·25-s + 4·27-s − 4·29-s − 12·39-s − 16·43-s − 2·49-s − 8·51-s − 12·53-s − 4·61-s + 32·69-s + 12·75-s + 5·81-s − 8·87-s + 20·101-s + 8·103-s + 40·107-s − 20·113-s − 18·117-s − 14·121-s + 127-s − 32·129-s + 131-s + ⋯
L(s)  = 1  + 1.15·3-s + 9-s − 1.66·13-s − 0.970·17-s + 3.33·23-s + 6/5·25-s + 0.769·27-s − 0.742·29-s − 1.92·39-s − 2.43·43-s − 2/7·49-s − 1.12·51-s − 1.64·53-s − 0.512·61-s + 3.85·69-s + 1.38·75-s + 5/9·81-s − 0.857·87-s + 1.99·101-s + 0.788·103-s + 3.86·107-s − 1.88·113-s − 1.66·117-s − 1.27·121-s + 0.0887·127-s − 2.81·129-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6230016\)    =    \(2^{12} \cdot 3^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(397.231\)
Root analytic conductor: \(4.46437\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6230016,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.128982668\)
\(L(\frac12)\) \(\approx\) \(3.128982668\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
13$C_2$ \( 1 + 6 T + p T^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.17.e_bm
19$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.19.a_abm
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.23.aq_eg
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.31.a_c
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.37.a_ak
41$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.41.a_ada
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.43.q_fu
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.47.a_acg
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \) 2.59.a_aek
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.67.a_aeo
71$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \) 2.71.a_aec
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.73.a_afa
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.83.a_be
89$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.89.a_afm
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.97.a_aby
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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

−9.299704927663673901006833605468, −8.830371090362162636185338103030, −8.527808366431448342238415959786, −7.88748949357421697573605624506, −7.61380568927496493543039070107, −7.15594867082694675668167122316, −7.02738892350570075061713603367, −6.49733663385318923406310370023, −6.31412157146577481102677899576, −5.23127915070968894527634640303, −5.14937127613438349746101876579, −4.63330446319798422676151403566, −4.61978334940188289228449376933, −3.69799005394653875336342838669, −3.21814880665933099978407766982, −2.96811714008796927765976969340, −2.56602693206352553943795233325, −1.91493673869620852990302477157, −1.45013655444023649261043720712, −0.53958494106187398616266466713, 0.53958494106187398616266466713, 1.45013655444023649261043720712, 1.91493673869620852990302477157, 2.56602693206352553943795233325, 2.96811714008796927765976969340, 3.21814880665933099978407766982, 3.69799005394653875336342838669, 4.61978334940188289228449376933, 4.63330446319798422676151403566, 5.14937127613438349746101876579, 5.23127915070968894527634640303, 6.31412157146577481102677899576, 6.49733663385318923406310370023, 7.02738892350570075061713603367, 7.15594867082694675668167122316, 7.61380568927496493543039070107, 7.88748949357421697573605624506, 8.527808366431448342238415959786, 8.830371090362162636185338103030, 9.299704927663673901006833605468

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