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Properties

Label 4-1620e2-1.1-c1e2-0-14
Degree $4$
Conductor $2624400$
Sign $1$
Analytic cond. $167.334$
Root an. cond. $3.59663$
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  + 5-s − 2·7-s − 2·13-s − 12·17-s − 8·19-s − 6·23-s − 6·29-s + 4·31-s − 2·35-s + 4·37-s − 6·41-s + 10·43-s + 6·47-s + 7·49-s − 12·53-s − 12·59-s − 2·61-s − 2·65-s − 2·67-s − 24·71-s + 4·73-s − 8·79-s − 6·83-s − 12·85-s − 12·89-s + 4·91-s − 8·95-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s − 0.554·13-s − 2.91·17-s − 1.83·19-s − 1.25·23-s − 1.11·29-s + 0.718·31-s − 0.338·35-s + 0.657·37-s − 0.937·41-s + 1.52·43-s + 0.875·47-s + 49-s − 1.64·53-s − 1.56·59-s − 0.256·61-s − 0.248·65-s − 0.244·67-s − 2.84·71-s + 0.468·73-s − 0.900·79-s − 0.658·83-s − 1.30·85-s − 1.27·89-s + 0.419·91-s − 0.820·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2624400\)    =    \(2^{4} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(167.334\)
Root analytic conductor: \(3.59663\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2624400,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3 \( 1 \)
5$C_2$ \( 1 - T + T^{2} \)
good7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_ad
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.11.a_al
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.c_aj
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.17.m_cs
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.23.g_n
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.29.g_h
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.ae_ap
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2^2$ \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_af
43$C_2^2$ \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.43.ak_cf
47$C_2^2$ \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.47.ag_al
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_dh
61$C_2^2$ \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.61.c_acf
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.67.c_acl
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.79.i_ap
83$C_2^2$ \( 1 + 6 T - 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.83.g_abv
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.97.c_adp
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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

−9.128268260075230639978810280679, −8.879609668633676560438179903739, −8.518701332677239511880295642056, −8.128006439029553491986246914026, −7.38160816732570281031336149602, −7.30574166128665362043917886705, −6.57545210475335878147183798094, −6.44791198724747539593326744519, −5.90277677478490330798174406784, −5.85851390667869065406076060278, −4.92223288312488907651471559014, −4.47514029994835495312362165577, −4.16105349651013886794664529191, −3.93944715685640249462758706971, −2.76324459854099422021748940251, −2.72807111839117201323872077617, −2.02989984762113154748685340570, −1.63169261711885065541736310622, 0, 0, 1.63169261711885065541736310622, 2.02989984762113154748685340570, 2.72807111839117201323872077617, 2.76324459854099422021748940251, 3.93944715685640249462758706971, 4.16105349651013886794664529191, 4.47514029994835495312362165577, 4.92223288312488907651471559014, 5.85851390667869065406076060278, 5.90277677478490330798174406784, 6.44791198724747539593326744519, 6.57545210475335878147183798094, 7.30574166128665362043917886705, 7.38160816732570281031336149602, 8.128006439029553491986246914026, 8.518701332677239511880295642056, 8.879609668633676560438179903739, 9.128268260075230639978810280679

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