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Properties

Label 4-132e2-1.1-c1e2-0-6
Degree $4$
Conductor $17424$
Sign $1$
Analytic cond. $1.11096$
Root an. cond. $1.02665$
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  + 3-s + 4·7-s − 2·9-s − 8·13-s + 16·19-s + 4·21-s − 25-s − 5·27-s + 10·31-s − 2·37-s − 8·39-s − 20·43-s − 2·49-s + 16·57-s − 8·61-s − 8·63-s − 2·67-s − 8·73-s − 75-s + 4·79-s + 81-s − 32·91-s + 10·93-s − 14·97-s + 16·103-s + 4·109-s − 2·111-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.51·7-s − 2/3·9-s − 2.21·13-s + 3.67·19-s + 0.872·21-s − 1/5·25-s − 0.962·27-s + 1.79·31-s − 0.328·37-s − 1.28·39-s − 3.04·43-s − 2/7·49-s + 2.11·57-s − 1.02·61-s − 1.00·63-s − 0.244·67-s − 0.936·73-s − 0.115·75-s + 0.450·79-s + 1/9·81-s − 3.35·91-s + 1.03·93-s − 1.42·97-s + 1.57·103-s + 0.383·109-s − 0.189·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(17424\)    =    \(2^{4} \cdot 3^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(1.11096\)
Root analytic conductor: \(1.02665\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 17424,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.418807030\)
\(L(\frac12)\) \(\approx\) \(1.418807030\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.7.ae_s
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.13.i_bq
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.19.aq_dy
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.23.a_bl
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.31.ak_dj
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.37.c_cx
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.43.u_he
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.59.a_ef
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.61.i_fi
67$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.67.c_ff
71$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.71.a_adf
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.73.i_gg
79$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.79.ae_gg
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.a_fa
89$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.89.a_dt
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.97.o_jj
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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

−11.33128539576622377975036396166, −10.21853205506356381906823281470, −9.813753565180213813561671136388, −9.542494902966876917365028895112, −8.721099137152928932255163369219, −8.132665807481574488057539990847, −7.60852684015633966980596452192, −7.46461921731656872852589989414, −6.54488810595431397607564592743, −5.38389169786521128568224402219, −5.11227132195889762323685903220, −4.66090457644079039135079955202, −3.27381985666804460223319703873, −2.80447621890231958099424449949, −1.61335599959865608201260450974, 1.61335599959865608201260450974, 2.80447621890231958099424449949, 3.27381985666804460223319703873, 4.66090457644079039135079955202, 5.11227132195889762323685903220, 5.38389169786521128568224402219, 6.54488810595431397607564592743, 7.46461921731656872852589989414, 7.60852684015633966980596452192, 8.132665807481574488057539990847, 8.721099137152928932255163369219, 9.542494902966876917365028895112, 9.813753565180213813561671136388, 10.21853205506356381906823281470, 11.33128539576622377975036396166

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