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Properties

Label 4-546e2-1.1-c1e2-0-45
Degree $4$
Conductor $298116$
Sign $1$
Analytic cond. $19.0081$
Root an. cond. $2.08802$
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  − 2-s + 3·3-s − 3·6-s + 5·7-s + 8-s + 6·9-s + 3·11-s + 7·13-s − 5·14-s − 16-s + 3·17-s − 6·18-s − 7·19-s + 15·21-s − 3·22-s − 12·23-s + 3·24-s − 2·25-s − 7·26-s + 9·27-s + 3·29-s − 8·31-s + 9·33-s − 3·34-s − 12·37-s + 7·38-s + 21·39-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.73·3-s − 1.22·6-s + 1.88·7-s + 0.353·8-s + 2·9-s + 0.904·11-s + 1.94·13-s − 1.33·14-s − 1/4·16-s + 0.727·17-s − 1.41·18-s − 1.60·19-s + 3.27·21-s − 0.639·22-s − 2.50·23-s + 0.612·24-s − 2/5·25-s − 1.37·26-s + 1.73·27-s + 0.557·29-s − 1.43·31-s + 1.56·33-s − 0.514·34-s − 1.97·37-s + 1.13·38-s + 3.36·39-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: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 298116,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.193935521\)
\(L(\frac12)\) \(\approx\) \(3.193935521\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
3$C_2$ \( 1 - p T + p T^{2} \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
13$C_2$ \( 1 - 7 T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.11.ad_ac
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.17.ad_ai
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.h_be
23$C_2^2$ \( 1 + 12 T + 71 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.23.m_ct
29$C_2^2$ \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.29.ad_bg
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2$ \( ( 1 + T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.37.m_dh
41$C_2^2$ \( 1 - 9 T + 68 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.41.aj_cq
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.43.ai_v
47$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \) 2.47.a_at
53$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \) 2.53.a_abf
59$C_2^2$ \( 1 + 18 T + 167 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.59.s_gl
61$C_2^2$ \( 1 - 9 T + 88 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.61.aj_dk
67$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.67.a_cp
71$C_2^2$ \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.71.g_abj
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.73.i_gg
79$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.79.w_kt
83$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.83.a_ba
89$C_2^2$ \( 1 - 15 T + 164 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.89.ap_gi
97$C_2^2$ \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.97.ac_adp
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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

−10.76404171652445162436368700314, −10.35755586450647557013418473890, −10.34770923065753354751382546040, −9.425874682483268862394938997865, −9.035291318085816790716224679813, −8.739937618297979873078353338759, −8.445119464798784168124227129641, −8.101814200092245320152805719619, −7.66825729244884065434916349889, −7.39024815444517330674212964731, −6.63660508751633759085438156484, −5.81676974811283150742554884998, −5.78103953968848611458012491794, −4.42096559079614079033351548551, −4.33438896394422983349627880181, −3.83325223241200571344385514153, −3.29562297744852499127160483122, −2.05434470899961253077337944268, −1.87452374099748778201388988223, −1.27799469535505028144983064005, 1.27799469535505028144983064005, 1.87452374099748778201388988223, 2.05434470899961253077337944268, 3.29562297744852499127160483122, 3.83325223241200571344385514153, 4.33438896394422983349627880181, 4.42096559079614079033351548551, 5.78103953968848611458012491794, 5.81676974811283150742554884998, 6.63660508751633759085438156484, 7.39024815444517330674212964731, 7.66825729244884065434916349889, 8.101814200092245320152805719619, 8.445119464798784168124227129641, 8.739937618297979873078353338759, 9.035291318085816790716224679813, 9.425874682483268862394938997865, 10.34770923065753354751382546040, 10.35755586450647557013418473890, 10.76404171652445162436368700314

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