↑

Properties

Label 4-4275e2-1.1-c1e2-0-5
Degree $4$
Conductor $18275625$
Sign $1$
Analytic cond. $1165.26$
Root an. cond. $5.84260$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4-s − 4·7-s − 8·13-s − 8·14-s + 16-s − 8·17-s + 2·19-s + 4·23-s − 16·26-s − 4·28-s + 4·29-s − 4·31-s − 2·32-s − 16·34-s − 8·37-s + 4·38-s + 12·41-s − 4·43-s + 8·46-s + 12·47-s − 8·52-s + 8·53-s + 8·58-s − 8·62-s − 11·64-s − 24·67-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s − 1.51·7-s − 2.21·13-s − 2.13·14-s + 1/4·16-s − 1.94·17-s + 0.458·19-s + 0.834·23-s − 3.13·26-s − 0.755·28-s + 0.742·29-s − 0.718·31-s − 0.353·32-s − 2.74·34-s − 1.31·37-s + 0.648·38-s + 1.87·41-s − 0.609·43-s + 1.17·46-s + 1.75·47-s − 1.10·52-s + 1.09·53-s + 1.05·58-s − 1.01·62-s − 1.37·64-s − 2.93·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18275625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(18275625\)    =    \(3^{4} \cdot 5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1165.26\)
Root analytic conductor: \(5.84260\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 18275625,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad3 \( 1 \)
5 \( 1 \)
19$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) 2.2.ac_d
7$D_{4}$ \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.7.e_q
11$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.11.a_u
13$D_{4}$ \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.13.i_bo
17$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.17.i_bq
23$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_s
29$D_{4}$ \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.29.ae_m
31$C_4$ \( 1 + 4 T - 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.31.e_ag
37$D_{4}$ \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.37.i_bo
41$D_{4}$ \( 1 - 12 T + 116 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.41.am_em
43$D_{4}$ \( 1 + 4 T + 88 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_dk
47$D_{4}$ \( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.47.am_du
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.53.ai_es
59$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.59.a_bu
61$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \) 2.61.a_dm
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.67.y_ks
71$D_{4}$ \( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.71.i_di
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.79.a_be
83$D_{4}$ \( 1 + 4 T + 98 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.83.e_du
89$D_{4}$ \( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.89.ae_u
97$C_2^2$ \( 1 + 176 T^{2} + p^{2} T^{4} \) 2.97.a_gu
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

−8.133914199365458465948147901354, −7.51115303589102797161468777280, −7.33414622823635359660799475023, −7.06815547519602159872205392482, −6.76978093544789675049433869471, −6.28748234578207812488485428215, −5.84170730043084026177542663517, −5.66043418127954607680151057209, −4.97596443820805829007042907843, −4.80374278906498192403494974829, −4.57995529020975816514391885431, −4.08175854079333487749314144381, −3.67315497872995440881707885943, −3.26432624673866600905196991037, −2.67751703285982928686704922668, −2.56481097968481408201878974001, −2.01799075059631958924524305554, −1.14446060622656722647525096372, 0, 0, 1.14446060622656722647525096372, 2.01799075059631958924524305554, 2.56481097968481408201878974001, 2.67751703285982928686704922668, 3.26432624673866600905196991037, 3.67315497872995440881707885943, 4.08175854079333487749314144381, 4.57995529020975816514391885431, 4.80374278906498192403494974829, 4.97596443820805829007042907843, 5.66043418127954607680151057209, 5.84170730043084026177542663517, 6.28748234578207812488485428215, 6.76978093544789675049433869471, 7.06815547519602159872205392482, 7.33414622823635359660799475023, 7.51115303589102797161468777280, 8.133914199365458465948147901354

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