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Properties

Label 4-1778112-1.1-c1e2-0-17
Degree $4$
Conductor $1778112$
Sign $-1$
Analytic cond. $113.373$
Root an. cond. $3.26308$
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 + 4-s − 7-s + 8-s − 4·11-s − 14-s + 16-s − 4·22-s + 8·23-s + 6·25-s − 28-s + 4·29-s + 32-s − 20·37-s − 12·43-s − 4·44-s + 8·46-s + 49-s + 6·50-s − 4·53-s − 56-s + 4·58-s + 64-s + 4·67-s − 20·74-s + 4·77-s + 8·79-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 1.20·11-s − 0.267·14-s + 1/4·16-s − 0.852·22-s + 1.66·23-s + 6/5·25-s − 0.188·28-s + 0.742·29-s + 0.176·32-s − 3.28·37-s − 1.82·43-s − 0.603·44-s + 1.17·46-s + 1/7·49-s + 0.848·50-s − 0.549·53-s − 0.133·56-s + 0.525·58-s + 1/8·64-s + 0.488·67-s − 2.32·74-s + 0.455·77-s + 0.900·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1778112 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1778112 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1778112\)    =    \(2^{6} \cdot 3^{4} \cdot 7^{3}\)
Sign: $-1$
Analytic conductor: \(113.373\)
Root analytic conductor: \(3.26308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1778112,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
7$C_1$ \( 1 + T \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
11$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.e_w
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.13.a_ag
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.19.a_aw
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.23.ai_bu
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.ae_bu
31$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.31.a_o
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.37.u_gs
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.a_as
43$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.43.m_di
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.e_bu
59$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.59.a_ag
61$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.61.a_k
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.ae_dy
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.a_da
73$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.73.a_abi
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.79.ai_gc
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.83.a_aw
89$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.89.a_abi
97$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \) 2.97.a_eg
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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

−7.50079486455555954715535670721, −6.97691300383953710655655456374, −6.70193141928706788753319140457, −6.54579638404460532631428676053, −5.71903976587663779229272962989, −5.22617719622723631613958176296, −5.02992090451124454001177669183, −4.78260510683382001760768018761, −3.94926308341524961919745044196, −3.37294431985496213272708453791, −3.08551966788659485422017339820, −2.62199725884763134242457022445, −1.88823954991993936773991656908, −1.15547407388376371069838846451, 0, 1.15547407388376371069838846451, 1.88823954991993936773991656908, 2.62199725884763134242457022445, 3.08551966788659485422017339820, 3.37294431985496213272708453791, 3.94926308341524961919745044196, 4.78260510683382001760768018761, 5.02992090451124454001177669183, 5.22617719622723631613958176296, 5.71903976587663779229272962989, 6.54579638404460532631428676053, 6.70193141928706788753319140457, 6.97691300383953710655655456374, 7.50079486455555954715535670721

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