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Properties

Label 4-450e2-1.1-c1e2-0-11
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 $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3·3-s + 3·6-s + 2·7-s − 8-s + 6·9-s + 3·11-s + 2·13-s + 2·14-s − 16-s + 6·17-s + 6·18-s − 2·19-s + 6·21-s + 3·22-s − 6·23-s − 3·24-s + 2·26-s + 9·27-s − 6·29-s + 4·31-s + 9·33-s + 6·34-s + 8·37-s − 2·38-s + 6·39-s − 9·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s + 1.22·6-s + 0.755·7-s − 0.353·8-s + 2·9-s + 0.904·11-s + 0.554·13-s + 0.534·14-s − 1/4·16-s + 1.45·17-s + 1.41·18-s − 0.458·19-s + 1.30·21-s + 0.639·22-s − 1.25·23-s − 0.612·24-s + 0.392·26-s + 1.73·27-s − 1.11·29-s + 0.718·31-s + 1.56·33-s + 1.02·34-s + 1.31·37-s − 0.324·38-s + 0.960·39-s − 1.40·41-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: \(0\)
Selberg data: \((4,\ 202500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.104353700\)
\(L(\frac12)\) \(\approx\) \(5.104353700\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_ad
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.11.ad_ac
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.13.ac_aj
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.17.ag_br
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.19.c_bn
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$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.37.ai_dm
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.41.j_bo
43$C_2^2$ \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) 2.43.b_abq
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.47.g_al
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.53.y_jq
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.59.d_aby
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.61.i_d
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.67.af_abq
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.73.w_kh
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.79.ae_acl
83$C_2^2$ \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_cj
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.af_acu
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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

−11.56771461296296119974295062547, −10.80194123723058709284773998801, −10.30396656391235994292315939764, −9.910997074477443606702770574616, −9.274872603280131576111493044147, −9.205433117171909534296475886627, −8.509200536342683308378546150141, −8.124245844746655401012290506779, −7.73600669066641568157569878012, −7.48734328902989086705339282572, −6.56908728749290567611420359529, −6.18520155344595074273102980887, −5.65412790301850388454428958179, −4.81895662114354649792334659128, −4.34543626391456700795034458123, −3.96066438097556818285336863425, −3.21032342138418895131192584177, −3.04404321685483644591487605851, −1.82189865160928531626798301504, −1.52995577731522410467364358928, 1.52995577731522410467364358928, 1.82189865160928531626798301504, 3.04404321685483644591487605851, 3.21032342138418895131192584177, 3.96066438097556818285336863425, 4.34543626391456700795034458123, 4.81895662114354649792334659128, 5.65412790301850388454428958179, 6.18520155344595074273102980887, 6.56908728749290567611420359529, 7.48734328902989086705339282572, 7.73600669066641568157569878012, 8.124245844746655401012290506779, 8.509200536342683308378546150141, 9.205433117171909534296475886627, 9.274872603280131576111493044147, 9.910997074477443606702770574616, 10.30396656391235994292315939764, 10.80194123723058709284773998801, 11.56771461296296119974295062547

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