↑

Properties

Label 4-311904-1.1-c1e2-0-19
Degree $4$
Conductor $311904$
Sign $1$
Analytic cond. $19.8872$
Root an. cond. $2.11175$
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 + 3-s + 4-s + 6·5-s − 6-s − 8-s + 9-s − 6·10-s + 12-s + 6·15-s + 16-s − 18-s − 2·19-s + 6·20-s − 24-s + 18·25-s + 27-s − 6·30-s + 6·31-s − 32-s + 36-s + 2·38-s − 6·40-s + 6·45-s + 48-s − 6·49-s − 18·50-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 2.68·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 1.89·10-s + 0.288·12-s + 1.54·15-s + 1/4·16-s − 0.235·18-s − 0.458·19-s + 1.34·20-s − 0.204·24-s + 18/5·25-s + 0.192·27-s − 1.09·30-s + 1.07·31-s − 0.176·32-s + 1/6·36-s + 0.324·38-s − 0.948·40-s + 0.894·45-s + 0.144·48-s − 6/7·49-s − 2.54·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(311904\)    =    \(2^{5} \cdot 3^{3} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(19.8872\)
Root analytic conductor: \(2.11175\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 311904,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.826106617\)
\(L(\frac12)\) \(\approx\) \(2.826106617\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
19$C_2$ \( 1 + 2 T + p T^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.5.ag_s
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.7.a_g
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
23$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.23.a_ba
29$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.29.a_abi
31$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.31.ag_ck
37$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.37.a_g
41$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.41.a_k
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.a_ao
47$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.47.a_ba
53$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.53.a_o
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.a_cc
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.67.ai_ek
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.a_da
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.a_eg
79$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.o_hq
83$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.83.a_aby
89$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.89.a_ade
97$C_2^2$ \( 1 + 174 T^{2} + p^{2} T^{4} \) 2.97.a_gs
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

−8.923268843103844861828455334607, −8.485922665308838727420865556669, −8.078966716202023698376208824441, −7.39457308942813865429562304774, −6.86705773814396300275345025345, −6.40022579185519670311395073938, −6.07041245574303060951569887560, −5.63809912523887288468777388959, −5.04364993723758411755085158766, −4.51079169084603022117774951723, −3.56524024166826191750674300170, −2.81509763399054230009310554993, −2.31066753227654707638037984112, −1.86151447951568220626122200470, −1.15985181956352118266231271206, 1.15985181956352118266231271206, 1.86151447951568220626122200470, 2.31066753227654707638037984112, 2.81509763399054230009310554993, 3.56524024166826191750674300170, 4.51079169084603022117774951723, 5.04364993723758411755085158766, 5.63809912523887288468777388959, 6.07041245574303060951569887560, 6.40022579185519670311395073938, 6.86705773814396300275345025345, 7.39457308942813865429562304774, 8.078966716202023698376208824441, 8.485922665308838727420865556669, 8.923268843103844861828455334607

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