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Properties

Label 4-1344e2-1.1-c1e2-0-31
Degree $4$
Conductor $1806336$
Sign $1$
Analytic cond. $115.173$
Root an. cond. $3.27595$
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 − 4·7-s + 3·9-s + 8·19-s − 8·21-s + 10·25-s + 4·27-s + 12·29-s + 8·31-s + 4·37-s + 9·49-s + 12·53-s + 16·57-s − 24·59-s − 12·63-s + 20·75-s + 5·81-s − 24·83-s + 24·87-s + 16·93-s + 8·103-s + 20·109-s + 8·111-s − 12·113-s + 10·121-s + 127-s + 131-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.51·7-s + 9-s + 1.83·19-s − 1.74·21-s + 2·25-s + 0.769·27-s + 2.22·29-s + 1.43·31-s + 0.657·37-s + 9/7·49-s + 1.64·53-s + 2.11·57-s − 3.12·59-s − 1.51·63-s + 2.30·75-s + 5/9·81-s − 2.63·83-s + 2.57·87-s + 1.65·93-s + 0.788·103-s + 1.91·109-s + 0.759·111-s − 1.12·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1806336\)    =    \(2^{12} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(115.173\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1806336,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.585288842\)
\(L(\frac12)\) \(\approx\) \(3.585288842\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.5.a_ak
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
13$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.13.a_aba
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.17.a_o
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.23.a_abi
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.41.a_abi
43$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.43.a_acw
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.59.y_kc
61$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.61.a_cs
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.a_aes
71$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.71.a_abi
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.a_bu
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.79.a_aby
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.83.y_ly
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.89.a_afa
97$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.97.a_ahm
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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

−9.685655751263249332963238057555, −9.568643256849823276027679515763, −8.864708976858723629115865537626, −8.689392078371401087606454770318, −8.397322514506065506757076552641, −7.76389343309206783142513694930, −7.32266435498185824988879637100, −7.09032917530441536565083679303, −6.54838997651263044106696991714, −6.26919411558892164731975585657, −5.76789264875777573192831545185, −5.09512361843874800711381336554, −4.49954553914434106491641236958, −4.40513414259135913602186157410, −3.34172226865863097934788458035, −3.24539867835608346879653980384, −2.81159075764828032651757567741, −2.47871467731910494702923127244, −1.27635779146320750506969557981, −0.837235830123698425905546974489, 0.837235830123698425905546974489, 1.27635779146320750506969557981, 2.47871467731910494702923127244, 2.81159075764828032651757567741, 3.24539867835608346879653980384, 3.34172226865863097934788458035, 4.40513414259135913602186157410, 4.49954553914434106491641236958, 5.09512361843874800711381336554, 5.76789264875777573192831545185, 6.26919411558892164731975585657, 6.54838997651263044106696991714, 7.09032917530441536565083679303, 7.32266435498185824988879637100, 7.76389343309206783142513694930, 8.397322514506065506757076552641, 8.689392078371401087606454770318, 8.864708976858723629115865537626, 9.568643256849823276027679515763, 9.685655751263249332963238057555

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