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Properties

Label 4-504e2-1.1-c1e2-0-32
Degree $4$
Conductor $254016$
Sign $-1$
Analytic cond. $16.1962$
Root an. cond. $2.00610$
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 − 2·4-s + 9-s + 2·12-s + 4·16-s − 9·17-s + 3·25-s − 27-s − 2·36-s + 18·41-s − 7·43-s − 4·48-s − 49-s + 9·51-s + 9·59-s − 8·64-s − 4·67-s + 18·68-s − 6·73-s − 3·75-s + 81-s + 18·83-s − 18·89-s − 16·97-s − 6·100-s + 2·108-s + 18·113-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s + 1/3·9-s + 0.577·12-s + 16-s − 2.18·17-s + 3/5·25-s − 0.192·27-s − 1/3·36-s + 2.81·41-s − 1.06·43-s − 0.577·48-s − 1/7·49-s + 1.26·51-s + 1.17·59-s − 64-s − 0.488·67-s + 2.18·68-s − 0.702·73-s − 0.346·75-s + 1/9·81-s + 1.97·83-s − 1.90·89-s − 1.62·97-s − 3/5·100-s + 0.192·108-s + 1.69·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(254016\)    =    \(2^{6} \cdot 3^{4} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(16.1962\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 254016,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + p T^{2} \)
3$C_1$ \( 1 + T \)
7$C_2$ \( 1 + T^{2} \)
good5$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.5.a_ad
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.a_s
13$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \) 2.13.a_x
17$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.17.j_bw
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.23.a_abj
29$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \) 2.29.a_d
31$C_2^2$ \( 1 + 29 T^{2} + p^{2} T^{4} \) 2.31.a_bd
37$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.37.a_f
41$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) 2.41.as_gd
43$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.h_da
47$C_2^2$ \( 1 + 33 T^{2} + p^{2} T^{4} \) 2.47.a_bh
53$C_2^2$ \( 1 + 43 T^{2} + p^{2} T^{4} \) 2.53.a_br
59$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.59.aj_fi
61$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.61.a_ak
67$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.67.e_cf
71$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.71.a_ad
73$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.73.g_cw
79$C_2^2$ \( 1 + 53 T^{2} + p^{2} T^{4} \) 2.79.a_cb
83$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.83.as_jn
89$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.89.s_jj
97$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.97.q_jy
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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

−8.776097988690363705510521607652, −8.411146929023139563887313698570, −7.71945110340183076726344561285, −7.33191975228528584983233725083, −6.65903920200006821523628212497, −6.34229108058848973253610270711, −5.75512529549239908125466922624, −5.17803693126665852420544776293, −4.72820061187515833814845344481, −4.20604854645190587026047000299, −3.88557714032528253183161985938, −2.89834932942318755102183466602, −2.22042848217764147519013469949, −1.12343461867709982681313058631, 0, 1.12343461867709982681313058631, 2.22042848217764147519013469949, 2.89834932942318755102183466602, 3.88557714032528253183161985938, 4.20604854645190587026047000299, 4.72820061187515833814845344481, 5.17803693126665852420544776293, 5.75512529549239908125466922624, 6.34229108058848973253610270711, 6.65903920200006821523628212497, 7.33191975228528584983233725083, 7.71945110340183076726344561285, 8.411146929023139563887313698570, 8.776097988690363705510521607652

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