↑

Properties

Label 4-194688-1.1-c1e2-0-1
Degree $4$
Conductor $194688$
Sign $1$
Analytic cond. $12.4134$
Root an. cond. $1.87703$
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-s + 4-s − 7-s − 8-s − 9-s + 14-s + 16-s − 2·17-s + 18-s − 2·23-s + 4·25-s − 28-s − 8·31-s − 32-s + 2·34-s − 36-s + 12·41-s + 2·46-s + 2·47-s − 49-s − 4·50-s + 56-s + 8·62-s + 63-s + 64-s − 2·68-s + 16·71-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 1/3·9-s + 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.417·23-s + 4/5·25-s − 0.188·28-s − 1.43·31-s − 0.176·32-s + 0.342·34-s − 1/6·36-s + 1.87·41-s + 0.294·46-s + 0.291·47-s − 1/7·49-s − 0.565·50-s + 0.133·56-s + 1.01·62-s + 0.125·63-s + 1/8·64-s − 0.242·68-s + 1.89·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 194688 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194688 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(194688\)    =    \(2^{7} \cdot 3^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(12.4134\)
Root analytic conductor: \(1.87703\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 194688,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9296656570\)
\(L(\frac12)\) \(\approx\) \(0.9296656570\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
13$C_2$ \( 1 + T^{2} \)
good5$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.5.a_ae
7$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.b_c
11$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.11.a_c
17$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.c_k
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$ \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.23.c_br
29$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.29.a_as
31$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.i_ck
37$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \) 2.37.a_abg
41$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.41.am_ek
43$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.43.a_k
47$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.47.ac_abx
53$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.53.a_q
59$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.59.a_ag
61$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \) 2.61.a_dm
67$C_2^2$ \( 1 + 96 T^{2} + p^{2} T^{4} \) 2.67.a_ds
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.71.aq_hi
73$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.73.m_fd
79$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) 2.79.ad_gc
83$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.83.a_aba
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + p T^{2} ) \) 2.89.ao_gw
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) 2.97.ay_mw
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

−9.181628232479985218068828780717, −8.821901675129735291499854865548, −8.097331287192322077548750579381, −7.80805304749904528104943026678, −7.16130358233683164438410451732, −6.88184270128025031714617742743, −6.08161155056658086746295730712, −5.95169227291797674635478009990, −5.16931197408396163152815510978, −4.58593645901433106364879790705, −3.83450269987269120824827831031, −3.27909177548567461258987474307, −2.51636032705536819884191917959, −1.89949973736680895786973015982, −0.68021622465261953620066118928, 0.68021622465261953620066118928, 1.89949973736680895786973015982, 2.51636032705536819884191917959, 3.27909177548567461258987474307, 3.83450269987269120824827831031, 4.58593645901433106364879790705, 5.16931197408396163152815510978, 5.95169227291797674635478009990, 6.08161155056658086746295730712, 6.88184270128025031714617742743, 7.16130358233683164438410451732, 7.80805304749904528104943026678, 8.097331287192322077548750579381, 8.821901675129735291499854865548, 9.181628232479985218068828780717

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