↑

Properties

Label 4-558e2-1.1-c1e2-0-2
Degree $4$
Conductor $311364$
Sign $1$
Analytic cond. $19.8528$
Root an. cond. $2.11084$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 3·5-s + 2·7-s − 4·8-s + 6·10-s + 11-s + 3·13-s − 4·14-s + 5·16-s + 17-s − 19-s − 9·20-s − 2·22-s + 16·23-s + 25-s − 6·26-s + 6·28-s + 6·29-s − 2·31-s − 6·32-s − 2·34-s − 6·35-s + 2·38-s + 12·40-s + 8·41-s − 10·43-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.34·5-s + 0.755·7-s − 1.41·8-s + 1.89·10-s + 0.301·11-s + 0.832·13-s − 1.06·14-s + 5/4·16-s + 0.242·17-s − 0.229·19-s − 2.01·20-s − 0.426·22-s + 3.33·23-s + 1/5·25-s − 1.17·26-s + 1.13·28-s + 1.11·29-s − 0.359·31-s − 1.06·32-s − 0.342·34-s − 1.01·35-s + 0.324·38-s + 1.89·40-s + 1.24·41-s − 1.52·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311364 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(311364\)    =    \(2^{2} \cdot 3^{4} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(19.8528\)
Root analytic conductor: \(2.11084\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 311364,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8840645809\)
\(L(\frac12)\) \(\approx\) \(0.8840645809\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
3 \( 1 \)
31$C_1$ \( ( 1 + T )^{2} \)
good5$C_2^2$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.5.d_i
7$D_{4}$ \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_ac
11$D_{4}$ \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) 2.11.ab_s
13$D_{4}$ \( 1 - 3 T + 24 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.13.ad_y
17$D_{4}$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) 2.17.ab_ae
19$D_{4}$ \( 1 + T + 34 T^{2} + p T^{3} + p^{2} T^{4} \) 2.19.b_bi
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.23.aq_eg
29$D_{4}$ \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.29.ag_by
37$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.37.a_g
41$D_{4}$ \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.41.ai_be
43$D_{4}$ \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.43.k_dq
47$D_{4}$ \( 1 + 5 T + 62 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.47.f_ck
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.53.ae_eg
59$D_{4}$ \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.59.g_eg
61$D_{4}$ \( 1 - 3 T + 120 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.61.ad_eq
67$D_{4}$ \( 1 - 13 T + 138 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.67.an_fi
71$D_{4}$ \( 1 - 7 T + 150 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.71.ah_fu
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.73.au_jm
79$D_{4}$ \( 1 - 5 T + 126 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.79.af_ew
83$D_{4}$ \( 1 + 5 T + 66 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.83.f_co
89$D_{4}$ \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.89.q_gs
97$D_{4}$ \( 1 + 9 T + 108 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.97.j_ee
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

−10.97912439437075354806411860759, −10.83467523404826639021220203717, −9.848469365548366722757573321205, −9.841125431601894532678622253037, −8.963181257994207900329427629596, −8.770020467204906007673979498594, −8.380997860273889822548624559224, −8.050715360883276528977195498424, −7.39483497183582346804446643075, −7.29264803347699720822342886900, −6.51314074706705780381261081986, −6.42683845425799820231756015098, −5.25836899397749777045352152287, −5.10805011182742789030922859556, −4.26263179516028591932408524640, −3.64227369316941992466762596184, −3.14140494476523898965119297517, −2.39487619836817467901745693046, −1.33299065917303741946874594155, −0.789388517802265969268664674090, 0.789388517802265969268664674090, 1.33299065917303741946874594155, 2.39487619836817467901745693046, 3.14140494476523898965119297517, 3.64227369316941992466762596184, 4.26263179516028591932408524640, 5.10805011182742789030922859556, 5.25836899397749777045352152287, 6.42683845425799820231756015098, 6.51314074706705780381261081986, 7.29264803347699720822342886900, 7.39483497183582346804446643075, 8.050715360883276528977195498424, 8.380997860273889822548624559224, 8.770020467204906007673979498594, 8.963181257994207900329427629596, 9.841125431601894532678622253037, 9.848469365548366722757573321205, 10.83467523404826639021220203717, 10.97912439437075354806411860759

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