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Properties

Label 4-76e4-1.1-c1e2-0-3
Degree $4$
Conductor $33362176$
Sign $1$
Analytic cond. $2127.20$
Root an. cond. $6.79128$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·5-s − 4·7-s − 4·9-s + 11·11-s − 2·13-s − 2·15-s + 10·17-s − 4·21-s − 5·23-s − 2·25-s − 6·27-s + 29-s + 11·31-s + 11·33-s + 8·35-s − 15·37-s − 2·39-s − 2·41-s − 43-s + 8·45-s − 12·47-s + 3·49-s + 10·51-s + 53-s − 22·55-s + 13·59-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s − 1.51·7-s − 4/3·9-s + 3.31·11-s − 0.554·13-s − 0.516·15-s + 2.42·17-s − 0.872·21-s − 1.04·23-s − 2/5·25-s − 1.15·27-s + 0.185·29-s + 1.97·31-s + 1.91·33-s + 1.35·35-s − 2.46·37-s − 0.320·39-s − 0.312·41-s − 0.152·43-s + 1.19·45-s − 1.75·47-s + 3/7·49-s + 1.40·51-s + 0.137·53-s − 2.96·55-s + 1.69·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33362176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33362176 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(33362176\)    =    \(2^{8} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(2127.20\)
Root analytic conductor: \(6.79128\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 33362176,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.730752895\)
\(L(\frac12)\) \(\approx\) \(2.730752895\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 \)
good3$D_{4}$ \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_f
5$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.5.c_g
7$D_{4}$ \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.7.e_n
11$C_4$ \( 1 - p T + 51 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) 2.11.al_bz
13$D_{4}$ \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.13.c_h
17$D_{4}$ \( 1 - 10 T + 54 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.17.ak_cc
23$D_{4}$ \( 1 + 5 T + 41 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.23.f_bp
29$D_{4}$ \( 1 - T + 47 T^{2} - p T^{3} + p^{2} T^{4} \) 2.29.ab_bv
31$C_4$ \( 1 - 11 T + 91 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.31.al_dn
37$D_{4}$ \( 1 + 15 T + 129 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.37.p_ez
41$D_{4}$ \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.41.c_d
43$D_{4}$ \( 1 + T + 85 T^{2} + p T^{3} + p^{2} T^{4} \) 2.43.b_dh
47$D_{4}$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.47.m_dh
53$D_{4}$ \( 1 - T + 95 T^{2} - p T^{3} + p^{2} T^{4} \) 2.53.ab_dr
59$D_{4}$ \( 1 - 13 T + 159 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.59.an_gd
61$C_2^2$ \( 1 + 77 T^{2} + p^{2} T^{4} \) 2.61.a_cz
67$C_2^2$ \( 1 + 89 T^{2} + p^{2} T^{4} \) 2.67.a_dl
71$D_{4}$ \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.71.ai_bh
73$D_{4}$ \( 1 - 20 T + 241 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.73.au_jh
79$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.79.am_hm
83$D_{4}$ \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_es
89$C_2^2$ \( 1 + 133 T^{2} + p^{2} T^{4} \) 2.89.a_fd
97$D_{4}$ \( 1 + 17 T + 205 T^{2} + 17 p T^{3} + p^{2} T^{4} \) 2.97.r_hx
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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

−8.279681701868830168797054766565, −8.216982973609517951178856670330, −7.69772393817729166402825695630, −7.02419819730126784505927377563, −6.82637883310086107246478595949, −6.66172494461054864273599653624, −6.09505023912583298414515649869, −6.07810986451570454695002643792, −5.39754422104235225030781536727, −5.16396357549855715465567228429, −4.50953974210629024807846354879, −3.90898778870189320354036443375, −3.78393777125075472783155626989, −3.46181705524464182698560446310, −3.13382494561391261233951004833, −2.97244079927548007476813659280, −1.86358858183971188358940607968, −1.85984389104259731366069536412, −0.791739815467697983454515623779, −0.57118606803571554812232021716, 0.57118606803571554812232021716, 0.791739815467697983454515623779, 1.85984389104259731366069536412, 1.86358858183971188358940607968, 2.97244079927548007476813659280, 3.13382494561391261233951004833, 3.46181705524464182698560446310, 3.78393777125075472783155626989, 3.90898778870189320354036443375, 4.50953974210629024807846354879, 5.16396357549855715465567228429, 5.39754422104235225030781536727, 6.07810986451570454695002643792, 6.09505023912583298414515649869, 6.66172494461054864273599653624, 6.82637883310086107246478595949, 7.02419819730126784505927377563, 7.69772393817729166402825695630, 8.216982973609517951178856670330, 8.279681701868830168797054766565

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