Properties

Label 4-136e2-1.1-c1e2-0-9
Degree $4$
Conductor $18496$
Sign $1$
Analytic cond. $1.17932$
Root an. cond. $1.04209$
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 + 4·5-s + 2·7-s + 2·9-s + 2·11-s − 8·15-s + 2·17-s − 4·19-s − 4·21-s + 2·23-s + 2·25-s − 6·27-s + 4·29-s − 2·31-s − 4·33-s + 8·35-s − 4·37-s + 4·41-s − 12·43-s + 8·45-s + 8·47-s − 6·49-s − 4·51-s − 4·53-s + 8·55-s + 8·57-s + 20·59-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.78·5-s + 0.755·7-s + 2/3·9-s + 0.603·11-s − 2.06·15-s + 0.485·17-s − 0.917·19-s − 0.872·21-s + 0.417·23-s + 2/5·25-s − 1.15·27-s + 0.742·29-s − 0.359·31-s − 0.696·33-s + 1.35·35-s − 0.657·37-s + 0.624·41-s − 1.82·43-s + 1.19·45-s + 1.16·47-s − 6/7·49-s − 0.560·51-s − 0.549·53-s + 1.07·55-s + 1.05·57-s + 2.60·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(18496\)    =    \(2^{6} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(1.17932\)
Root analytic conductor: \(1.04209\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 18496,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.137059284\)
\(L(\frac12)\) \(\approx\) \(1.137059284\)
\(L(\frac{3}{2})\) 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 \( 1 \)
17$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_c
5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.5.ae_o
7$D_{4}$ \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_k
11$D_{4}$ \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_s
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
19$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_w
23$D_{4}$ \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.23.ac_bq
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.29.ae_ck
31$D_{4}$ \( 1 + 2 T + 58 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.31.c_cg
37$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.37.e_ac
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$D_{4}$ \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.43.m_dy
47$D_{4}$ \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.47.ai_be
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.53.e_eg
59$D_{4}$ \( 1 - 20 T + 198 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.59.au_hq
61$D_{4}$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.61.e_bu
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.67.y_ks
71$D_{4}$ \( 1 - 14 T + 186 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.71.ao_he
73$D_{4}$ \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.73.am_dy
79$D_{4}$ \( 1 - 10 T + 138 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.79.ak_fi
83$D_{4}$ \( 1 - 12 T + 182 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_ha
89$D_{4}$ \( 1 + 24 T + 302 T^{2} + 24 p T^{3} + p^{2} T^{4} \) 2.89.y_lq
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−13.49085759577834040588555307572, −12.95499863335649123679834245047, −12.44836080142256483393647294834, −11.88176029212561145998688564212, −11.40819645886218929442505664158, −10.99546039656749745576780395701, −10.29935346640717734907909684785, −10.05473493622270074119771615729, −9.440210581016239559833347742153, −8.942534572435227892849471776987, −8.213281007835140825652083695990, −7.54227009460585271658092107954, −6.55803287402778873278660169255, −6.44153995685194095508342264679, −5.59734670055123138599998759290, −5.37445701147005362698510239113, −4.62453500205073604857732525057, −3.69329493216429982385966860300, −2.24827141653266557516415079165, −1.50989576196998004253019603835, 1.50989576196998004253019603835, 2.24827141653266557516415079165, 3.69329493216429982385966860300, 4.62453500205073604857732525057, 5.37445701147005362698510239113, 5.59734670055123138599998759290, 6.44153995685194095508342264679, 6.55803287402778873278660169255, 7.54227009460585271658092107954, 8.213281007835140825652083695990, 8.942534572435227892849471776987, 9.440210581016239559833347742153, 10.05473493622270074119771615729, 10.29935346640717734907909684785, 10.99546039656749745576780395701, 11.40819645886218929442505664158, 11.88176029212561145998688564212, 12.44836080142256483393647294834, 12.95499863335649123679834245047, 13.49085759577834040588555307572

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