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Properties

Label 4-132e2-1.1-c1e2-0-5
Degree $4$
Conductor $17424$
Sign $1$
Analytic cond. $1.11096$
Root an. cond. $1.02665$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2·3-s + 4-s − 2·6-s − 8-s + 9-s + 2·11-s + 2·12-s + 16-s + 8·17-s − 18-s − 2·22-s − 2·24-s − 4·25-s − 4·27-s − 12·29-s − 32-s + 4·33-s − 8·34-s + 36-s + 4·37-s + 4·41-s + 2·44-s + 2·48-s − 8·49-s + 4·50-s + 16·51-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.353·8-s + 1/3·9-s + 0.603·11-s + 0.577·12-s + 1/4·16-s + 1.94·17-s − 0.235·18-s − 0.426·22-s − 0.408·24-s − 4/5·25-s − 0.769·27-s − 2.22·29-s − 0.176·32-s + 0.696·33-s − 1.37·34-s + 1/6·36-s + 0.657·37-s + 0.624·41-s + 0.301·44-s + 0.288·48-s − 8/7·49-s + 0.565·50-s + 2.24·51-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: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 17424,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.196531640\)
\(L(\frac12)\) \(\approx\) \(1.196531640\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
3$C_2$ \( 1 - 2 T + p T^{2} \)
11$C_2$ \( 1 - 2 T + p T^{2} \)
good5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.5.a_e
7$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.7.a_i
13$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.13.a_e
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.17.ai_by
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2^2$ \( 1 - 24 T^{2} + p^{2} T^{4} \) 2.23.a_ay
29$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.m_da
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.ae_o
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.ae_w
43$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.43.a_s
47$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \) 2.47.a_abo
53$C_2^2$ \( 1 + 68 T^{2} + p^{2} T^{4} \) 2.53.a_cq
59$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.59.a_acs
61$C_2^2$ \( 1 - 76 T^{2} + p^{2} T^{4} \) 2.61.a_acy
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.67.aq_ha
71$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.71.a_i
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.a_eg
79$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.79.a_i
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.a_as
97$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.97.ai_fq
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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

−11.03898724040794083964176833851, −10.18889371515235895772000647808, −9.660510485698010670502508183082, −9.435884446263282402630148899748, −8.904790231241950527152938848997, −8.141332301349384762782929533829, −7.79064638618317172298217062679, −7.44847926954597412473763049644, −6.57421179015927283499116930979, −5.81416886483910222173817349793, −5.28936089704830193113594589170, −3.91553939482346923496127857358, −3.54372529731937322199342356953, −2.59295342718213135246868120382, −1.56871314210694090717909164327, 1.56871314210694090717909164327, 2.59295342718213135246868120382, 3.54372529731937322199342356953, 3.91553939482346923496127857358, 5.28936089704830193113594589170, 5.81416886483910222173817349793, 6.57421179015927283499116930979, 7.44847926954597412473763049644, 7.79064638618317172298217062679, 8.141332301349384762782929533829, 8.904790231241950527152938848997, 9.435884446263282402630148899748, 9.660510485698010670502508183082, 10.18889371515235895772000647808, 11.03898724040794083964176833851

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