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Properties

Label 4-107424-1.1-c1e2-0-0
Degree $4$
Conductor $107424$
Sign $-1$
Analytic cond. $6.84944$
Root an. cond. $1.61775$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

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 − 3·11-s − 2·12-s − 2·13-s + 16-s − 18-s + 3·22-s − 3·23-s + 2·24-s + 8·25-s + 2·26-s + 4·27-s − 32-s + 6·33-s + 36-s + 4·37-s + 4·39-s − 3·44-s + 3·46-s + 9·47-s − 2·48-s − 13·49-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 − 0.904·11-s − 0.577·12-s − 0.554·13-s + 1/4·16-s − 0.235·18-s + 0.639·22-s − 0.625·23-s + 0.408·24-s + 8/5·25-s + 0.392·26-s + 0.769·27-s − 0.176·32-s + 1.04·33-s + 1/6·36-s + 0.657·37-s + 0.640·39-s − 0.452·44-s + 0.442·46-s + 1.31·47-s − 0.288·48-s − 1.85·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(107424\)    =    \(2^{5} \cdot 3^{2} \cdot 373\)
Sign: $-1$
Analytic conductor: \(6.84944\)
Root analytic conductor: \(1.61775\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 107424,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 2 T + p T^{2} \)
373$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
good5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.5.a_ai
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.a_n
11$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.d_e
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.13.c_bb
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.17.a_ao
19$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.19.a_au
23$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.d_bc
29$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \) 2.29.a_z
31$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \) 2.31.a_abj
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2^2$ \( 1 - 71 T^{2} + p^{2} T^{4} \) 2.41.a_act
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.43.a_aby
47$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.47.aj_ei
53$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.53.a_ac
59$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) 2.59.ad_eo
61$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.61.ah_y
67$C_2^2$ \( 1 + 100 T^{2} + p^{2} T^{4} \) 2.67.a_dw
71$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) 2.71.ad_fm
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.ae_g
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.a_cg
83$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.83.s_id
89$C_2^2$ \( 1 - 53 T^{2} + p^{2} T^{4} \) 2.89.a_acb
97$C_2$$\times$$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) 2.97.bd_ou
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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

−9.503163954561047175484445302631, −8.645995560525923641944901820003, −8.369142587991770008440678816270, −7.82304824165043592531558379970, −7.22009359164400080066910921121, −6.77474484112761121557576039011, −6.32502525571489596900834203305, −5.63494030548728239730056930660, −5.27204206640441048842905968280, −4.73382303819021367088333451864, −3.99688278426096601185469222075, −2.92535345412322664233390963865, −2.46589927347859519522376038231, −1.20463346804124289440373458390, 0, 1.20463346804124289440373458390, 2.46589927347859519522376038231, 2.92535345412322664233390963865, 3.99688278426096601185469222075, 4.73382303819021367088333451864, 5.27204206640441048842905968280, 5.63494030548728239730056930660, 6.32502525571489596900834203305, 6.77474484112761121557576039011, 7.22009359164400080066910921121, 7.82304824165043592531558379970, 8.369142587991770008440678816270, 8.645995560525923641944901820003, 9.503163954561047175484445302631

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