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Properties

Label 4-9680e2-1.1-c1e2-0-3
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  − 3-s − 2·5-s + 3·7-s − 9-s − 2·13-s + 2·15-s − 7·17-s + 9·19-s − 3·21-s − 6·23-s + 3·25-s + 5·29-s − 5·31-s − 6·35-s − 37-s + 2·39-s + 20·41-s + 2·43-s + 2·45-s + 6·47-s − 3·49-s + 7·51-s − 13·53-s − 9·57-s + 2·59-s + 3·61-s − 3·63-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 1.13·7-s − 1/3·9-s − 0.554·13-s + 0.516·15-s − 1.69·17-s + 2.06·19-s − 0.654·21-s − 1.25·23-s + 3/5·25-s + 0.928·29-s − 0.898·31-s − 1.01·35-s − 0.164·37-s + 0.320·39-s + 3.12·41-s + 0.304·43-s + 0.298·45-s + 0.875·47-s − 3/7·49-s + 0.980·51-s − 1.78·53-s − 1.19·57-s + 0.260·59-s + 0.384·61-s − 0.377·63-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

\(L(1)\) \(\approx\) \(1.698915351\)
\(L(\frac12)\) \(\approx\) \(1.698915351\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) 2.3.b_c
7$D_{4}$ \( 1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.7.ad_m
13$C_4$ \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.13.c_k
17$D_{4}$ \( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.17.h_bq
19$D_{4}$ \( 1 - 9 T + 54 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.19.aj_cc
23$D_{4}$ \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.23.g_bm
29$D_{4}$ \( 1 - 5 T + 60 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.29.af_ci
31$D_{4}$ \( 1 + 5 T + 30 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.31.f_be
37$D_{4}$ \( 1 + T - 32 T^{2} + p T^{3} + p^{2} T^{4} \) 2.37.b_abg
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.41.au_ha
43$D_{4}$ \( 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.43.ac_cs
47$D_{4}$ \( 1 - 6 T + 86 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.47.ag_di
53$D_{4}$ \( 1 + 13 T + 144 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.53.n_fo
59$D_{4}$ \( 1 - 2 T - 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.59.ac_abi
61$D_{4}$ \( 1 - 3 T + 120 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.61.ad_eq
67$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.67.a_fe
71$D_{4}$ \( 1 - 9 T + 158 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.71.aj_gc
73$D_{4}$ \( 1 + 18 T + 210 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.73.s_ic
79$D_{4}$ \( 1 - 10 T + 166 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.79.ak_gk
83$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.83.au_kg
89$D_{4}$ \( 1 + T + 72 T^{2} + p T^{3} + p^{2} T^{4} \) 2.89.b_cu
97$D_{4}$ \( 1 - 26 T + 346 T^{2} - 26 p T^{3} + p^{2} T^{4} \) 2.97.aba_ni
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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

−7.71757776877214733658824471252, −7.48592897308533257220503072844, −7.41610614001904489919409690438, −6.91514263576475299516286763907, −6.34617082475969521303939301253, −6.21717514072273057621136537537, −5.70220538958208522503390394776, −5.57350002110435635509318167587, −4.83328659809043794188609148431, −4.73666520752694850221683402486, −4.65969024370817705739210529722, −4.07204203530136266060364144131, −3.57380954395202073141426774481, −3.43444997252985248498818343085, −2.69108842147570682530367953226, −2.39422471113254328376395343192, −2.00538506219869449645960073331, −1.41137435669781449557502064742, −0.78608357438396380565795427270, −0.40932763667768038606561334184, 0.40932763667768038606561334184, 0.78608357438396380565795427270, 1.41137435669781449557502064742, 2.00538506219869449645960073331, 2.39422471113254328376395343192, 2.69108842147570682530367953226, 3.43444997252985248498818343085, 3.57380954395202073141426774481, 4.07204203530136266060364144131, 4.65969024370817705739210529722, 4.73666520752694850221683402486, 4.83328659809043794188609148431, 5.57350002110435635509318167587, 5.70220538958208522503390394776, 6.21717514072273057621136537537, 6.34617082475969521303939301253, 6.91514263576475299516286763907, 7.41610614001904489919409690438, 7.48592897308533257220503072844, 7.71757776877214733658824471252

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