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Properties

Label 4-756e2-1.1-c1e2-0-19
Degree $4$
Conductor $571536$
Sign $1$
Analytic cond. $36.4416$
Root an. cond. $2.45696$
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  + 6·5-s + 4·7-s − 12·17-s + 17·25-s + 24·35-s + 2·37-s − 6·41-s + 20·43-s − 12·47-s + 9·49-s + 12·59-s + 4·67-s + 28·79-s + 12·83-s − 72·85-s + 18·89-s + 36·101-s − 22·109-s − 48·119-s − 5·121-s + 18·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  + 2.68·5-s + 1.51·7-s − 2.91·17-s + 17/5·25-s + 4.05·35-s + 0.328·37-s − 0.937·41-s + 3.04·43-s − 1.75·47-s + 9/7·49-s + 1.56·59-s + 0.488·67-s + 3.15·79-s + 1.31·83-s − 7.80·85-s + 1.90·89-s + 3.58·101-s − 2.10·109-s − 4.40·119-s − 0.454·121-s + 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(571536\)    =    \(2^{4} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(36.4416\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 571536,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.892768911\)
\(L(\frac12)\) \(\approx\) \(3.892768911\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.5.ag_t
11$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.11.a_f
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.13.a_ao
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.17.m_cs
19$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \) 2.19.a_abj
23$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \) 2.23.a_at
29$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.29.a_by
31$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \) 2.31.a_abj
37$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.37.ac_cx
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.41.g_dn
43$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.43.au_he
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.47.m_fa
53$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.53.a_aec
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.59.am_fy
61$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.61.a_cs
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.67.ae_fi
71$C_2^2$ \( 1 - 115 T^{2} + p^{2} T^{4} \) 2.71.a_ael
73$C_2^2$ \( 1 - 134 T^{2} + p^{2} T^{4} \) 2.73.a_afe
79$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.79.abc_nq
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.83.am_hu
89$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.89.as_jz
97$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \) 2.97.a_afq
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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

−10.42496749245610712327378364349, −10.32568702344136244060203219218, −9.528051441513835824394691167351, −9.284623571176793489351541801442, −8.990673177544833188893339377022, −8.660084583323062691237669680778, −7.952537059707057697467903010867, −7.67606099440537164763174787528, −6.74501712182444563765675206637, −6.60749595961290575475892651466, −6.16478919125706584032999629009, −5.71135370211101716666465681475, −5.03837005126779033082726554224, −4.95729342157511820235378866601, −4.36652821479421606476640961845, −3.64674112267195386288722672565, −2.40930699420365285803489465925, −2.21162985299984596235187084605, −1.99298582318335137164776507068, −1.07219492169125979655317125175, 1.07219492169125979655317125175, 1.99298582318335137164776507068, 2.21162985299984596235187084605, 2.40930699420365285803489465925, 3.64674112267195386288722672565, 4.36652821479421606476640961845, 4.95729342157511820235378866601, 5.03837005126779033082726554224, 5.71135370211101716666465681475, 6.16478919125706584032999629009, 6.60749595961290575475892651466, 6.74501712182444563765675206637, 7.67606099440537164763174787528, 7.952537059707057697467903010867, 8.660084583323062691237669680778, 8.990673177544833188893339377022, 9.284623571176793489351541801442, 9.528051441513835824394691167351, 10.32568702344136244060203219218, 10.42496749245610712327378364349

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