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Properties

Label 4-3888e2-1.1-c1e2-0-7
Degree $4$
Conductor $15116544$
Sign $1$
Analytic cond. $963.843$
Root an. cond. $5.57187$
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  − 5-s + 3·7-s − 8·11-s + 2·13-s − 2·17-s − 19-s + 7·23-s − 25-s + 9·29-s + 11·31-s − 3·35-s − 9·37-s − 6·41-s + 7·43-s − 15·47-s + 49-s + 17·53-s + 8·55-s − 8·59-s + 5·61-s − 2·65-s + 8·67-s + 5·71-s + 2·73-s − 24·77-s + 4·79-s + 2·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.13·7-s − 2.41·11-s + 0.554·13-s − 0.485·17-s − 0.229·19-s + 1.45·23-s − 1/5·25-s + 1.67·29-s + 1.97·31-s − 0.507·35-s − 1.47·37-s − 0.937·41-s + 1.06·43-s − 2.18·47-s + 1/7·49-s + 2.33·53-s + 1.07·55-s − 1.04·59-s + 0.640·61-s − 0.248·65-s + 0.977·67-s + 0.593·71-s + 0.234·73-s − 2.73·77-s + 0.450·79-s + 0.219·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15116544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15116544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(15116544\)    =    \(2^{8} \cdot 3^{10}\)
Sign: $1$
Analytic conductor: \(963.843\)
Root analytic conductor: \(5.57187\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 15116544,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.557518555\)
\(L(\frac12)\) \(\approx\) \(2.557518555\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_c
7$D_{4}$ \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.7.ad_i
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.11.i_bm
13$D_{4}$ \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.13.ac_ag
17$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.17.c_c
19$D_{4}$ \( 1 + T + 30 T^{2} + p T^{3} + p^{2} T^{4} \) 2.19.b_be
23$D_{4}$ \( 1 - 7 T + 50 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.23.ah_by
29$D_{4}$ \( 1 - 9 T + 70 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.29.aj_cs
31$D_{4}$ \( 1 - 11 T + 84 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.31.al_dg
37$D_{4}$ \( 1 + 9 T + 86 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.37.j_di
41$D_{4}$ \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_cg
43$D_{4}$ \( 1 - 7 T + 90 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.43.ah_dm
47$D_{4}$ \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.47.p_fm
53$D_{4}$ \( 1 - 17 T + 170 T^{2} - 17 p T^{3} + p^{2} T^{4} \) 2.53.ar_go
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$D_{4}$ \( 1 - 5 T + 120 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.61.af_eq
67$D_{4}$ \( 1 - 8 T + 117 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_en
71$D_{4}$ \( 1 - 5 T + 74 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.71.af_cw
73$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.73.ac_fr
79$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.79.ae_be
83$D_{4}$ \( 1 - 2 T + 134 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.83.ac_fe
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.89.ay_mk
97$D_{4}$ \( 1 - 13 T + 228 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.97.an_iu
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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

−8.480766726803733953368794900411, −8.455277793345612872279255271037, −7.888846280462354815344771673510, −7.66927233073932585433915220459, −7.38972241993387254395217101416, −6.79156513440065187565930422136, −6.39571716960358292396130095748, −6.28789558406316010538134751947, −5.40314149433498952426544557921, −5.09095990260703691075230695664, −5.07728109767811729301974751155, −4.60238360285427010750523709214, −4.25792305691166914373057088648, −3.52345324409897028053804989326, −3.12672603001372809445376837137, −2.80171615332915873172819004271, −2.15820470945753519047378059442, −1.92787308369069984543900282859, −0.937290039658782137150622796630, −0.57071804065959510841952202243, 0.57071804065959510841952202243, 0.937290039658782137150622796630, 1.92787308369069984543900282859, 2.15820470945753519047378059442, 2.80171615332915873172819004271, 3.12672603001372809445376837137, 3.52345324409897028053804989326, 4.25792305691166914373057088648, 4.60238360285427010750523709214, 5.07728109767811729301974751155, 5.09095990260703691075230695664, 5.40314149433498952426544557921, 6.28789558406316010538134751947, 6.39571716960358292396130095748, 6.79156513440065187565930422136, 7.38972241993387254395217101416, 7.66927233073932585433915220459, 7.888846280462354815344771673510, 8.455277793345612872279255271037, 8.480766726803733953368794900411

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