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Properties

Label 4-2366e2-1.1-c1e2-0-13
Degree $4$
Conductor $5597956$
Sign $1$
Analytic cond. $356.930$
Root an. cond. $4.34656$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 2·3-s + 3·4-s − 2·5-s − 4·6-s − 2·7-s + 4·8-s − 4·10-s + 2·11-s − 6·12-s − 4·14-s + 4·15-s + 5·16-s − 8·17-s − 4·19-s − 6·20-s + 4·21-s + 4·22-s + 6·23-s − 8·24-s − 7·25-s + 2·27-s − 6·28-s − 6·29-s + 8·30-s + 14·31-s + 6·32-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s − 0.755·7-s + 1.41·8-s − 1.26·10-s + 0.603·11-s − 1.73·12-s − 1.06·14-s + 1.03·15-s + 5/4·16-s − 1.94·17-s − 0.917·19-s − 1.34·20-s + 0.872·21-s + 0.852·22-s + 1.25·23-s − 1.63·24-s − 7/5·25-s + 0.384·27-s − 1.13·28-s − 1.11·29-s + 1.46·30-s + 2.51·31-s + 1.06·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5597956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5597956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5597956\)    =    \(2^{2} \cdot 7^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(356.930\)
Root analytic conductor: \(4.34656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 5597956,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
13 \( 1 \)
good3$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_e
5$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.5.c_l
11$D_{4}$ \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_u
17$C_2^2$ \( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.17.i_bv
19$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_be
23$D_{4}$ \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_ca
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.29.g_cp
31$D_{4}$ \( 1 - 14 T + 108 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.31.ao_ee
37$C_2^2$ \( 1 + 47 T^{2} + p^{2} T^{4} \) 2.37.a_bv
41$D_{4}$ \( 1 - 2 T + 71 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.41.ac_ct
43$D_{4}$ \( 1 + 14 T + 108 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.43.o_ee
47$D_{4}$ \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.47.i_ck
53$D_{4}$ \( 1 - 10 T + 119 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.53.ak_ep
59$D_{4}$ \( 1 + 18 T + 196 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.59.s_ho
61$D_{4}$ \( 1 + 20 T + 219 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.61.u_il
67$D_{4}$ \( 1 - 6 T + 68 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.67.ag_cq
71$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.71.a_aby
73$D_{4}$ \( 1 + 2 T + 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_bn
79$D_{4}$ \( 1 + 18 T + 212 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.79.s_ie
83$D_{4}$ \( 1 + 18 T + 220 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.83.s_im
89$D_{4}$ \( 1 + 12 T + 202 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_hu
97$D_{4}$ \( 1 - 4 T + 186 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.97.ae_he
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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

−8.547594992164205336936871408753, −8.515555040556149667672946354483, −7.84023186710153715636150745632, −7.48368614926966973344361268034, −6.94889281087645164076829857500, −6.60783682001252835112777770542, −6.30453265784210065031363530924, −6.25021980417363742317247007268, −5.51018590872417861098917411555, −5.36956929014345924273026435746, −4.52245195760403103912932975837, −4.46011350816031448948214450035, −4.17882185693419083832418332538, −3.59619903384691236174980648320, −2.91204849639759265843545105556, −2.85452964335640877013887870029, −1.93566937048094380354569967793, −1.39103844041953203269856385468, 0, 0, 1.39103844041953203269856385468, 1.93566937048094380354569967793, 2.85452964335640877013887870029, 2.91204849639759265843545105556, 3.59619903384691236174980648320, 4.17882185693419083832418332538, 4.46011350816031448948214450035, 4.52245195760403103912932975837, 5.36956929014345924273026435746, 5.51018590872417861098917411555, 6.25021980417363742317247007268, 6.30453265784210065031363530924, 6.60783682001252835112777770542, 6.94889281087645164076829857500, 7.48368614926966973344361268034, 7.84023186710153715636150745632, 8.515555040556149667672946354483, 8.547594992164205336936871408753

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