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Properties

Label 4-546e2-1.1-c1e2-0-41
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4-s + 7-s − 2·9-s + 12-s − 5·13-s + 16-s + 9·19-s − 21-s + 25-s + 5·27-s − 28-s + 2·36-s − 12·37-s + 5·39-s + 7·43-s − 48-s − 6·49-s + 5·52-s − 9·57-s − 7·61-s − 2·63-s − 64-s − 12·67-s − 3·73-s − 75-s − 9·76-s + ⋯
L(s)  = 1  − 0.577·3-s − 1/2·4-s + 0.377·7-s − 2/3·9-s + 0.288·12-s − 1.38·13-s + 1/4·16-s + 2.06·19-s − 0.218·21-s + 1/5·25-s + 0.962·27-s − 0.188·28-s + 1/3·36-s − 1.97·37-s + 0.800·39-s + 1.06·43-s − 0.144·48-s − 6/7·49-s + 0.693·52-s − 1.19·57-s − 0.896·61-s − 0.251·63-s − 1/8·64-s − 1.46·67-s − 0.351·73-s − 0.115·75-s − 1.03·76-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: \(1\)
Selberg data: \((4,\ 298116,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T + p T^{2} \)
7$C_2$ \( 1 - T + p T^{2} \)
13$C_2$ \( 1 + 5 T + p T^{2} \)
good5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.5.a_ab
11$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.11.a_an
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.17.a_k
19$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.19.aj_cg
23$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.23.a_bi
29$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.29.a_b
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.31.a_bl
37$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.37.m_ef
41$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \) 2.41.a_bj
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.43.ah_da
47$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \) 2.47.a_abl
53$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \) 2.53.a_abj
59$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.59.a_ak
61$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.61.h_fc
67$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.67.m_gn
71$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \) 2.71.a_bj
73$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.d_ai
79$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.79.k_fr
83$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \) 2.83.a_es
89$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.89.a_ck
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.97.aj_hc
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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

−8.686836190715184505127885609093, −8.104226781334655880930739296202, −7.55152430199211692002423762878, −7.33473662621060057601176881095, −6.77295697427469563403916013078, −6.09473282972611192416641183059, −5.46207567148366667837571545516, −5.35182095006627685856391217041, −4.72361526819582109684734549146, −4.37623146707406366783849450095, −3.23950154977748907776759428969, −3.14061781719437279344983796951, −2.14601114415903561742151833183, −1.16300789867369033865478670026, 0, 1.16300789867369033865478670026, 2.14601114415903561742151833183, 3.14061781719437279344983796951, 3.23950154977748907776759428969, 4.37623146707406366783849450095, 4.72361526819582109684734549146, 5.35182095006627685856391217041, 5.46207567148366667837571545516, 6.09473282972611192416641183059, 6.77295697427469563403916013078, 7.33473662621060057601176881095, 7.55152430199211692002423762878, 8.104226781334655880930739296202, 8.686836190715184505127885609093

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