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Properties

Label 4-546e2-1.1-c1e2-0-30
Degree $4$
Conductor $298116$
Sign $1$
Analytic cond. $19.0081$
Root an. cond. $2.08802$
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  + 3-s − 4-s − 2·9-s − 12-s + 6·13-s + 16-s + 12·19-s + 9·25-s − 5·27-s + 4·31-s + 2·36-s − 2·37-s + 6·39-s − 12·43-s + 48-s + 49-s − 6·52-s + 12·57-s − 5·61-s − 64-s − 20·67-s − 12·73-s + 9·75-s − 12·76-s + 10·79-s + 81-s + 4·93-s + ⋯
L(s)  = 1  + 0.577·3-s − 1/2·4-s − 2/3·9-s − 0.288·12-s + 1.66·13-s + 1/4·16-s + 2.75·19-s + 9/5·25-s − 0.962·27-s + 0.718·31-s + 1/3·36-s − 0.328·37-s + 0.960·39-s − 1.82·43-s + 0.144·48-s + 1/7·49-s − 0.832·52-s + 1.58·57-s − 0.640·61-s − 1/8·64-s − 2.44·67-s − 1.40·73-s + 1.03·75-s − 1.37·76-s + 1.12·79-s + 1/9·81-s + 0.414·93-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(298116\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(19.0081\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 298116,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.165291390\)
\(L(\frac12)\) \(\approx\) \(2.165291390\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 - T + p T^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_2$ \( 1 - 6 T + p T^{2} \)
good5$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \) 2.5.a_aj
11$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.11.a_af
17$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.17.a_r
19$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.19.am_cv
23$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \) 2.23.a_ar
29$C_2^2$ \( 1 - 41 T^{2} + p^{2} T^{4} \) 2.29.a_abp
31$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.31.ae_by
37$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.c_cw
41$C_2^2$ \( 1 - 15 T^{2} + p^{2} T^{4} \) 2.41.a_ap
43$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.m_eo
47$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.47.a_f
53$C_2^2$ \( 1 - 101 T^{2} + p^{2} T^{4} \) 2.53.a_adx
59$C_2^2$ \( 1 + 105 T^{2} + p^{2} T^{4} \) 2.59.a_eb
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.61.f_ee
67$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.u_hq
71$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.71.a_h
73$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.m_de
79$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.79.ak_fr
83$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \) 2.83.a_aeg
89$C_2^2$ \( 1 + 85 T^{2} + p^{2} T^{4} \) 2.89.a_dh
97$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 - 12 T + p T^{2} ) \) 2.97.abe_pu
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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

−8.844462594342556686413760357438, −8.541345073161162991865921191685, −7.901970281342507801265299952741, −7.63387805545414500658670818644, −7.01502842228840028247741179989, −6.43557342179424431059040572881, −5.93304684443894674433660571832, −5.42936613812177500958205378805, −4.96299607546879585210397813322, −4.42236058316872853140001339718, −3.48374024140300898680999189779, −3.26419279322283984276892223134, −2.87780032541234489013395254365, −1.62250923930147037930720994646, −0.927917143785814056993105056714, 0.927917143785814056993105056714, 1.62250923930147037930720994646, 2.87780032541234489013395254365, 3.26419279322283984276892223134, 3.48374024140300898680999189779, 4.42236058316872853140001339718, 4.96299607546879585210397813322, 5.42936613812177500958205378805, 5.93304684443894674433660571832, 6.43557342179424431059040572881, 7.01502842228840028247741179989, 7.63387805545414500658670818644, 7.901970281342507801265299952741, 8.541345073161162991865921191685, 8.844462594342556686413760357438

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