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Properties

Label 4-3888e2-1.1-c1e2-0-9
Degree $4$
Conductor $15116544$
Sign $1$
Analytic cond. $963.843$
Root an. cond. $5.57187$
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  + 14·13-s + 10·25-s − 20·37-s + 2·49-s − 2·61-s + 14·73-s − 10·97-s + 38·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 121·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 3.88·13-s + 2·25-s − 3.28·37-s + 2/7·49-s − 0.256·61-s + 1.63·73-s − 1.01·97-s + 3.63·109-s − 2·121-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 + 0.0783·163-s + 0.0773·167-s + 9.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15116544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15116544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(15116544\)    =    \(2^{8} \cdot 3^{10}\)
Sign: $1$
Analytic conductor: \(963.843\)
Root analytic conductor: \(5.57187\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 15116544,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.748139217\)
\(L(\frac12)\) \(\approx\) \(3.748139217\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.5.a_ak
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.13.ao_cx
17$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.17.a_abi
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.19.a_bl
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.29.a_acg
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.31.a_ach
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.37.u_gs
41$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.41.a_ade
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.43.a_cj
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.53.a_aec
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.61.c_et
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.67.a_ef
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.73.ao_hn
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.79.a_al
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.89.a_agw
97$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.97.k_il
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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

−8.648173569120783048371797116863, −8.363775097511202739903551185434, −8.225039821818100512882014905555, −7.49208945147544445925587808219, −7.14691369607926644055245484746, −6.75525817159696980087210638883, −6.33143306465351778195295893059, −6.24624752711333909673115492040, −5.76592348594305163753504720222, −5.17148531094982666507873303392, −5.11732544082023343873175539295, −4.44371299682345400071587370038, −3.83722939288064617642930650453, −3.66206420187866528855571909752, −3.34902963488700180398630672735, −2.91504781655165375126994778097, −2.14546156595631139977257221352, −1.47350296796455746691979770460, −1.29026934868701591123432602584, −0.62164740539072338227831102203, 0.62164740539072338227831102203, 1.29026934868701591123432602584, 1.47350296796455746691979770460, 2.14546156595631139977257221352, 2.91504781655165375126994778097, 3.34902963488700180398630672735, 3.66206420187866528855571909752, 3.83722939288064617642930650453, 4.44371299682345400071587370038, 5.11732544082023343873175539295, 5.17148531094982666507873303392, 5.76592348594305163753504720222, 6.24624752711333909673115492040, 6.33143306465351778195295893059, 6.75525817159696980087210638883, 7.14691369607926644055245484746, 7.49208945147544445925587808219, 8.225039821818100512882014905555, 8.363775097511202739903551185434, 8.648173569120783048371797116863

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