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Properties

Label 4-363e2-1.1-c1e2-0-6
Degree $4$
Conductor $131769$
Sign $1$
Analytic cond. $8.40170$
Root an. cond. $1.70251$
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  + 2-s + 2·3-s − 2·4-s − 3·5-s + 2·6-s + 6·7-s − 3·8-s + 3·9-s − 3·10-s − 4·12-s + 8·13-s + 6·14-s − 6·15-s + 16-s + 17-s + 3·18-s + 5·19-s + 6·20-s + 12·21-s − 2·23-s − 6·24-s − 2·25-s + 8·26-s + 4·27-s − 12·28-s − 6·30-s − 31-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s − 4-s − 1.34·5-s + 0.816·6-s + 2.26·7-s − 1.06·8-s + 9-s − 0.948·10-s − 1.15·12-s + 2.21·13-s + 1.60·14-s − 1.54·15-s + 1/4·16-s + 0.242·17-s + 0.707·18-s + 1.14·19-s + 1.34·20-s + 2.61·21-s − 0.417·23-s − 1.22·24-s − 2/5·25-s + 1.56·26-s + 0.769·27-s − 2.26·28-s − 1.09·30-s − 0.179·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(131769\)    =    \(3^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(8.40170\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 131769,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.901707972\)
\(L(\frac12)\) \(\approx\) \(2.901707972\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3$C_1$ \( ( 1 - T )^{2} \)
11 \( 1 \)
good2$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) 2.2.ab_d
5$D_{4}$ \( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.5.d_l
7$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.7.ag_x
13$D_{4}$ \( 1 - 8 T + 37 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.13.ai_bl
17$D_{4}$ \( 1 - T + 33 T^{2} - p T^{3} + p^{2} T^{4} \) 2.17.ab_bh
19$D_{4}$ \( 1 - 5 T + 33 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.19.af_bh
23$D_{4}$ \( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_bb
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.29.a_bm
31$D_{4}$ \( 1 + T + 51 T^{2} + p T^{3} + p^{2} T^{4} \) 2.31.b_bz
37$D_{4}$ \( 1 + 4 T + 73 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.37.e_cv
41$D_{4}$ \( 1 + 6 T + 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_l
43$D_{4}$ \( 1 - 8 T + 97 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.43.ai_dt
47$D_{4}$ \( 1 - T + 93 T^{2} - p T^{3} + p^{2} T^{4} \) 2.47.ab_dp
53$D_{4}$ \( 1 + 17 T + 177 T^{2} + 17 p T^{3} + p^{2} T^{4} \) 2.53.r_gv
59$D_{4}$ \( 1 - 5 T + 63 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.59.af_cl
61$D_{4}$ \( 1 - 9 T + 131 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.61.aj_fb
67$D_{4}$ \( 1 - T + 33 T^{2} - p T^{3} + p^{2} T^{4} \) 2.67.ab_bh
71$D_{4}$ \( 1 - 9 T + 61 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.71.aj_cj
73$D_{4}$ \( 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_fm
79$D_{4}$ \( 1 - 10 T + 163 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.79.ak_gh
83$D_{4}$ \( 1 + 12 T + 157 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_gb
89$D_{4}$ \( 1 - 10 T + 183 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.89.ak_hb
97$D_{4}$ \( 1 - T - 17 T^{2} - p T^{3} + p^{2} T^{4} \) 2.97.ab_ar
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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

−11.50398749116343667739074270250, −11.42839163980825173977288130985, −10.89581250907366373252466324033, −10.44187354639346064627124619476, −9.439131134488799967977287715585, −9.406146736517919592281054524453, −8.513070950631042094322441331278, −8.396210739694550870956355430360, −8.030904019089371728678372939174, −7.79526601890347810468062167552, −7.16710424225055155957688020296, −6.33312158733603880861521065816, −5.41772466495137110744896176397, −5.23332069893214574001633976046, −4.35999116674849621885614761016, −4.21280858197488992738118398970, −3.58177656039584076646750252854, −3.32008581174686335844406129445, −1.92517922199044961747531354354, −1.18710578751622857863542613928, 1.18710578751622857863542613928, 1.92517922199044961747531354354, 3.32008581174686335844406129445, 3.58177656039584076646750252854, 4.21280858197488992738118398970, 4.35999116674849621885614761016, 5.23332069893214574001633976046, 5.41772466495137110744896176397, 6.33312158733603880861521065816, 7.16710424225055155957688020296, 7.79526601890347810468062167552, 8.030904019089371728678372939174, 8.396210739694550870956355430360, 8.513070950631042094322441331278, 9.406146736517919592281054524453, 9.439131134488799967977287715585, 10.44187354639346064627124619476, 10.89581250907366373252466324033, 11.42839163980825173977288130985, 11.50398749116343667739074270250

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