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Properties

Label 4-133e2-1.1-c1e2-0-4
Degree $4$
Conductor $17689$
Sign $1$
Analytic cond. $1.12786$
Root an. cond. $1.03053$
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 + 3·3-s − 2·4-s + 2·5-s + 3·6-s + 2·7-s − 3·8-s + 2·9-s + 2·10-s − 11-s − 6·12-s − 2·13-s + 2·14-s + 6·15-s + 16-s + 17-s + 2·18-s − 2·19-s − 4·20-s + 6·21-s − 22-s − 2·23-s − 9·24-s − 7·25-s − 2·26-s − 6·27-s − 4·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s − 4-s + 0.894·5-s + 1.22·6-s + 0.755·7-s − 1.06·8-s + 2/3·9-s + 0.632·10-s − 0.301·11-s − 1.73·12-s − 0.554·13-s + 0.534·14-s + 1.54·15-s + 1/4·16-s + 0.242·17-s + 0.471·18-s − 0.458·19-s − 0.894·20-s + 1.30·21-s − 0.213·22-s − 0.417·23-s − 1.83·24-s − 7/5·25-s − 0.392·26-s − 1.15·27-s − 0.755·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17689 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(17689\)    =    \(7^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1.12786\)
Root analytic conductor: \(1.03053\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 17689,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.226521401\)
\(L(\frac12)\) \(\approx\) \(2.226521401\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad7$C_1$ \( ( 1 - T )^{2} \)
19$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) 2.2.ab_d
3$D_{4}$ \( 1 - p T + 7 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) 2.3.ad_h
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.5.ac_l
11$C_4$ \( 1 + T + 21 T^{2} + p T^{3} + p^{2} T^{4} \) 2.11.b_v
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.13.c_bb
17$D_{4}$ \( 1 - T + 23 T^{2} - p T^{3} + p^{2} T^{4} \) 2.17.ab_x
23$D_{4}$ \( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_bb
29$D_{4}$ \( 1 - 5 T + 63 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.29.af_cl
31$C_4$ \( 1 + T - 39 T^{2} + p T^{3} + p^{2} T^{4} \) 2.31.b_abn
37$D_{4}$ \( 1 + 14 T + 103 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.37.o_dz
41$C_4$ \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.41.aj_ct
43$D_{4}$ \( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.43.ai_de
47$D_{4}$ \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.47.ag_df
53$D_{4}$ \( 1 - 3 T + 97 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.53.ad_dt
59$D_{4}$ \( 1 - 20 T + 213 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.59.au_if
61$D_{4}$ \( 1 + 6 T + 51 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.61.g_bz
67$D_{4}$ \( 1 - 11 T + 103 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.67.al_dz
71$D_{4}$ \( 1 - 4 T + 101 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.71.ae_dx
73$D_{4}$ \( 1 + 7 T + 97 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.73.h_dt
79$C_2^2$ \( 1 + 138 T^{2} + p^{2} T^{4} \) 2.79.a_fi
83$D_{4}$ \( 1 - 13 T + 197 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.83.an_hp
89$D_{4}$ \( 1 - 10 T + 198 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.89.ak_hq
97$D_{4}$ \( 1 - 6 T + 183 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.97.ag_hb
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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

−13.62025060895511532214578037311, −13.45486519438234806445427367698, −12.71041944691866674160718347910, −12.25370563588666584498674988050, −11.71265111051365992000621233065, −10.85844929122760418066354542346, −10.22529099332162626706503366805, −9.766895861616560528521506473258, −9.053331509949530621405899261300, −8.994154382011203061738546809188, −8.262282164725696725643339386455, −7.936094927570823561051604323227, −7.25895705220832828808564338575, −6.20826058264757330489467877805, −5.40571077549911655311343714364, −5.19434758733118900016763246405, −4.05900994656425477202054392423, −3.79392358318710283826542288661, −2.62047472811705218653323646518, −2.12384877547305281217510389717, 2.12384877547305281217510389717, 2.62047472811705218653323646518, 3.79392358318710283826542288661, 4.05900994656425477202054392423, 5.19434758733118900016763246405, 5.40571077549911655311343714364, 6.20826058264757330489467877805, 7.25895705220832828808564338575, 7.936094927570823561051604323227, 8.262282164725696725643339386455, 8.994154382011203061738546809188, 9.053331509949530621405899261300, 9.766895861616560528521506473258, 10.22529099332162626706503366805, 10.85844929122760418066354542346, 11.71265111051365992000621233065, 12.25370563588666584498674988050, 12.71041944691866674160718347910, 13.45486519438234806445427367698, 13.62025060895511532214578037311

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