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Properties

Label 4-450e2-1.1-c1e2-0-12
Degree $4$
Conductor $202500$
Sign $1$
Analytic cond. $12.9115$
Root an. cond. $1.89559$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3·3-s + 3·6-s − 4·7-s + 8-s + 6·9-s − 3·11-s − 4·13-s + 4·14-s − 16-s + 6·17-s − 6·18-s − 8·19-s + 12·21-s + 3·22-s − 6·23-s − 3·24-s + 4·26-s − 9·27-s + 6·29-s − 8·31-s + 9·33-s − 6·34-s − 16·37-s + 8·38-s + 12·39-s + 6·41-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s + 1.22·6-s − 1.51·7-s + 0.353·8-s + 2·9-s − 0.904·11-s − 1.10·13-s + 1.06·14-s − 1/4·16-s + 1.45·17-s − 1.41·18-s − 1.83·19-s + 2.61·21-s + 0.639·22-s − 1.25·23-s − 0.612·24-s + 0.784·26-s − 1.73·27-s + 1.11·29-s − 1.43·31-s + 1.56·33-s − 1.02·34-s − 2.63·37-s + 1.29·38-s + 1.92·39-s + 0.937·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(202500\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(12.9115\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 202500,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + T + T^{2} \)
3$C_2$ \( 1 + p T + p T^{2} \)
5 \( 1 \)
good7$C_2$ \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.7.e_j
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_ac
13$C_2^2$ \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.13.e_d
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.17.ag_br
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.ag_h
31$C_2^2$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.31.i_bh
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.37.q_fi
41$C_2^2$ \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_af
43$C_2^2$ \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) 2.43.b_abq
47$C_2^2$ \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.47.am_dt
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2^2$ \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.59.aj_w
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.61.i_d
67$C_2^2$ \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_abz
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.71.m_gw
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.73.bc_ne
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.79.i_ap
83$C_2^2$ \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.83.j_ac
89$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.89.s_jz
97$C_2^2$ \( 1 + 7 T - 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.97.h_abw
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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

−10.54911184061948457325943695062, −10.41929159624427283601670106434, −10.05049605378469152312790453051, −9.909530277399859716106180001218, −9.144004254066264603035661950833, −8.703415464149144452575373545354, −8.139305372682385888948115746165, −7.45385497477481971104160936022, −6.98160606473388361582558565131, −6.93152735287150688178153848895, −5.94731165742957080282610511362, −5.73465248675527525835266042215, −5.45367246275482720453311730346, −4.45620166140926607087639742337, −4.26574334934088432481035944222, −3.30859819364722614495584263813, −2.54315119257022247051796027279, −1.53972805685923414681871814262, 0, 0, 1.53972805685923414681871814262, 2.54315119257022247051796027279, 3.30859819364722614495584263813, 4.26574334934088432481035944222, 4.45620166140926607087639742337, 5.45367246275482720453311730346, 5.73465248675527525835266042215, 5.94731165742957080282610511362, 6.93152735287150688178153848895, 6.98160606473388361582558565131, 7.45385497477481971104160936022, 8.139305372682385888948115746165, 8.703415464149144452575373545354, 9.144004254066264603035661950833, 9.909530277399859716106180001218, 10.05049605378469152312790453051, 10.41929159624427283601670106434, 10.54911184061948457325943695062

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