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Properties

Label 4-1778112-1.1-c1e2-0-4
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s + 6·11-s + 2·23-s + 6·25-s − 4·29-s + 4·37-s − 12·43-s + 49-s − 8·53-s + 24·67-s − 10·71-s − 6·77-s − 10·107-s + 4·109-s − 12·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 2·161-s + 163-s + 167-s + 18·169-s + ⋯
L(s)  = 1  − 0.377·7-s + 1.80·11-s + 0.417·23-s + 6/5·25-s − 0.742·29-s + 0.657·37-s − 1.82·43-s + 1/7·49-s − 1.09·53-s + 2.93·67-s − 1.18·71-s − 0.683·77-s − 0.966·107-s + 0.383·109-s − 1.12·113-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s − 0.157·161-s + 0.0783·163-s + 0.0773·167-s + 1.38·169-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: \(0\)
Selberg data: \((4,\ 1778112,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.322709788\)
\(L(\frac12)\) \(\approx\) \(2.322709788\)
\(L(\frac{3}{2})\) 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 \( 1 \)
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$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.11.ag_be
13$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.13.a_as
17$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.17.a_ag
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.a_aba
23$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.23.ac_bu
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.e_cg
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.31.a_ba
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ae_ck
41$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.41.a_abe
43$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.m_eo
47$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.47.a_ag
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.53.i_w
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.a_cc
61$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \) 2.61.a_dq
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.67.ay_ks
71$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.71.k_di
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.a_bu
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.83.a_acc
89$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.89.a_ady
97$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.97.a_bm
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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

−7.910489147431906635124545293093, −7.16206904495078102443340603759, −6.95823931859025270937304818580, −6.51567766184194342113532039110, −6.32378599902051787370213083891, −5.66148254005099040758837895576, −5.20647332325804300823585378129, −4.74070507335651174958467615835, −4.16397640153808834363449325538, −3.81603535652702936958928874958, −3.23033010258396086200303787533, −2.86875159974836367637069146908, −1.95799433761455871462119607011, −1.44679592177937758802165601442, −0.67247478348687725196487606645, 0.67247478348687725196487606645, 1.44679592177937758802165601442, 1.95799433761455871462119607011, 2.86875159974836367637069146908, 3.23033010258396086200303787533, 3.81603535652702936958928874958, 4.16397640153808834363449325538, 4.74070507335651174958467615835, 5.20647332325804300823585378129, 5.66148254005099040758837895576, 6.32378599902051787370213083891, 6.51567766184194342113532039110, 6.95823931859025270937304818580, 7.16206904495078102443340603759, 7.910489147431906635124545293093

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