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Properties

Label 4-9680e2-1.1-c1e2-0-1
Degree $4$
Conductor $93702400$
Sign $1$
Analytic cond. $5974.54$
Root an. cond. $8.79176$
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 − 2·5-s − 3·7-s + 2·9-s − 13-s + 4·15-s − 6·17-s − 3·19-s + 6·21-s − 7·23-s + 3·25-s − 6·27-s + 2·29-s + 10·31-s + 6·35-s + 5·37-s + 2·39-s + 3·41-s + 14·43-s − 4·45-s − 7·47-s − 6·49-s + 12·51-s + 13·53-s + 6·57-s − 7·59-s − 6·63-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s − 1.13·7-s + 2/3·9-s − 0.277·13-s + 1.03·15-s − 1.45·17-s − 0.688·19-s + 1.30·21-s − 1.45·23-s + 3/5·25-s − 1.15·27-s + 0.371·29-s + 1.79·31-s + 1.01·35-s + 0.821·37-s + 0.320·39-s + 0.468·41-s + 2.13·43-s − 0.596·45-s − 1.02·47-s − 6/7·49-s + 1.68·51-s + 1.78·53-s + 0.794·57-s − 0.911·59-s − 0.755·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93702400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93702400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(93702400\)    =    \(2^{8} \cdot 5^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(5974.54\)
Root analytic conductor: \(8.79176\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 93702400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5582559201\)
\(L(\frac12)\) \(\approx\) \(0.5582559201\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
11 \( 1 \)
good3$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_c
7$D_{4}$ \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.7.d_p
13$D_{4}$ \( 1 + T + 25 T^{2} + p T^{3} + p^{2} T^{4} \) 2.13.b_z
17$D_{4}$ \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.17.g_bm
19$D_{4}$ \( 1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.19.d_j
23$D_{4}$ \( 1 + 7 T + 47 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.23.h_bv
29$D_{4}$ \( 1 - 2 T + 54 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.29.ac_cc
31$D_{4}$ \( 1 - 10 T + 82 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.31.ak_de
37$D_{4}$ \( 1 - 5 T + 69 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.37.af_cr
41$D_{4}$ \( 1 - 3 T + 73 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.41.ad_cv
43$D_{4}$ \( 1 - 14 T + 130 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.43.ao_fa
47$D_{4}$ \( 1 + 7 T + 45 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.47.h_bt
53$D_{4}$ \( 1 - 13 T + 117 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.53.an_en
59$D_{4}$ \( 1 + 7 T + 129 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.59.h_ez
61$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \) 2.61.a_bq
67$D_{4}$ \( 1 + 6 T + 138 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.67.g_fi
71$D_{4}$ \( 1 + 22 T + 258 T^{2} + 22 p T^{3} + p^{2} T^{4} \) 2.71.w_jy
73$D_{4}$ \( 1 - 10 T + 126 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.73.ak_ew
79$D_{4}$ \( 1 - 12 T + 174 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.79.am_gs
83$D_{4}$ \( 1 + 18 T + 202 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.83.s_hu
89$D_{4}$ \( 1 - 5 T - 27 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.89.af_abb
97$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.97.y_na
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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

−7.894041900003607680192439493765, −7.21172718762720095041813107045, −7.12037534030346037843643935690, −6.99524804507811846095618856368, −6.21912941720026742424929655854, −6.11967973081497228135437846900, −5.96729495272783120044679637259, −5.80184518387821585927331736034, −4.87662409705550519742368812303, −4.64610907227142247750983758100, −4.48773502683268198742048876290, −4.16967067508240579086604271544, −3.61855096008773108682041569148, −3.41658277652481328852569102095, −2.62420925270325583092328655702, −2.57241130377826461363688696251, −1.99370046213876412241361982658, −1.34185997323568125631143307886, −0.56191008818192870719012523499, −0.32879311922149039520095520022, 0.32879311922149039520095520022, 0.56191008818192870719012523499, 1.34185997323568125631143307886, 1.99370046213876412241361982658, 2.57241130377826461363688696251, 2.62420925270325583092328655702, 3.41658277652481328852569102095, 3.61855096008773108682041569148, 4.16967067508240579086604271544, 4.48773502683268198742048876290, 4.64610907227142247750983758100, 4.87662409705550519742368812303, 5.80184518387821585927331736034, 5.96729495272783120044679637259, 6.11967973081497228135437846900, 6.21912941720026742424929655854, 6.99524804507811846095618856368, 7.12037534030346037843643935690, 7.21172718762720095041813107045, 7.894041900003607680192439493765

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