↑

Properties

Label 4-46144-1.1-c1e2-0-0
Degree $4$
Conductor $46144$
Sign $1$
Analytic cond. $2.94218$
Root an. cond. $1.30968$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 9-s + 4·16-s − 3·17-s + 3·23-s − 25-s + 10·31-s − 2·36-s + 15·41-s + 3·47-s − 6·49-s − 8·64-s + 6·68-s + 15·71-s + 22·73-s − 20·79-s − 8·81-s + 18·89-s − 6·92-s − 20·97-s + 2·100-s − 3·103-s + 3·113-s + 14·121-s − 20·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 4-s + 1/3·9-s + 16-s − 0.727·17-s + 0.625·23-s − 1/5·25-s + 1.79·31-s − 1/3·36-s + 2.34·41-s + 0.437·47-s − 6/7·49-s − 64-s + 0.727·68-s + 1.78·71-s + 2.57·73-s − 2.25·79-s − 8/9·81-s + 1.90·89-s − 0.625·92-s − 2.03·97-s + 1/5·100-s − 0.295·103-s + 0.282·113-s + 1.27·121-s − 1.79·124-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 46144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(46144\)    =    \(2^{6} \cdot 7 \cdot 103\)
Sign: $1$
Analytic conductor: \(2.94218\)
Root analytic conductor: \(1.30968\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 46144,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.061274016\)
\(L(\frac12)\) \(\approx\) \(1.061274016\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
7$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + T + p T^{2} ) \)
103$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
good3$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.3.a_ab
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.a_ao
13$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.13.a_ai
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.d_bi
19$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.19.a_au
23$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) 2.23.ad_bu
29$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.29.a_ax
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.31.ak_dj
37$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.37.a_au
41$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.41.ap_fg
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.a_ao
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.47.ad_ao
53$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.53.a_bi
59$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.59.a_bu
61$C_2^2$ \( 1 + 112 T^{2} + p^{2} T^{4} \) 2.61.a_ei
67$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.67.a_ai
71$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + p T^{2} ) \) 2.71.ap_fm
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.73.aw_kh
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.79.u_jy
83$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.83.a_bo
89$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.89.as_jq
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_li
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.954130819029835794819604171427, −9.630255654806861465656282591121, −9.246571996874011317785879619249, −8.584298554986868987179559559321, −8.197221316652313522358736352910, −7.67636597512318422651042636876, −6.99399480284247447212186323547, −6.40833707877687508651430902704, −5.80599597096701852138562962796, −5.11996417192076159449236963079, −4.49219533117301553261282005959, −4.12512390743888765996693136408, −3.23137238852655407297325754736, −2.35747539104312951687887317033, −0.958265763869207414551715350530, 0.958265763869207414551715350530, 2.35747539104312951687887317033, 3.23137238852655407297325754736, 4.12512390743888765996693136408, 4.49219533117301553261282005959, 5.11996417192076159449236963079, 5.80599597096701852138562962796, 6.40833707877687508651430902704, 6.99399480284247447212186323547, 7.67636597512318422651042636876, 8.197221316652313522358736352910, 8.584298554986868987179559559321, 9.246571996874011317785879619249, 9.630255654806861465656282591121, 9.954130819029835794819604171427

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