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Properties

Label 4-405e2-1.1-c1e2-0-5
Degree $4$
Conductor $164025$
Sign $-1$
Analytic cond. $10.4583$
Root an. cond. $1.79831$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s + 4·7-s − 8·13-s + 12·16-s − 2·19-s + 25-s − 16·28-s − 2·31-s + 16·37-s − 8·43-s − 2·49-s + 32·52-s − 20·61-s − 32·64-s + 28·67-s + 4·73-s + 8·76-s − 32·79-s − 32·91-s − 8·97-s − 4·100-s + 28·103-s − 2·109-s + 48·112-s − 13·121-s + 8·124-s + 127-s + ⋯
L(s)  = 1  − 2·4-s + 1.51·7-s − 2.21·13-s + 3·16-s − 0.458·19-s + 1/5·25-s − 3.02·28-s − 0.359·31-s + 2.63·37-s − 1.21·43-s − 2/7·49-s + 4.43·52-s − 2.56·61-s − 4·64-s + 3.42·67-s + 0.468·73-s + 0.917·76-s − 3.60·79-s − 3.35·91-s − 0.812·97-s − 2/5·100-s + 2.75·103-s − 0.191·109-s + 4.53·112-s − 1.18·121-s + 0.718·124-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(164025\)    =    \(3^{8} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(10.4583\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 164025,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.2.a_e
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.7.ae_s
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.11.a_n
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.13.i_bq
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.19.c_bn
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.29.a_ax
31$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.31.c_cl
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.37.aq_fi
41$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.41.a_cv
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.59.a_ef
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.67.abc_ms
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.71.a_fd
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.79.bg_py
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.89.a_abv
97$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.97.i_ic
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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

−8.922941609669290152495843868069, −8.538687880420415041094117740367, −8.027105501047314182413259178583, −7.73102144096255023104365276788, −7.36532580760380206733382868882, −6.47289908140934830552102870516, −5.75188631797463493014423176019, −5.18690236062207152876422818521, −4.76552433542106688375778775229, −4.59634975877424853850281958730, −4.04017076909632160868815003758, −3.14233849758258863442381605357, −2.30189805881882976209364768483, −1.27645214139714047950709921081, 0, 1.27645214139714047950709921081, 2.30189805881882976209364768483, 3.14233849758258863442381605357, 4.04017076909632160868815003758, 4.59634975877424853850281958730, 4.76552433542106688375778775229, 5.18690236062207152876422818521, 5.75188631797463493014423176019, 6.47289908140934830552102870516, 7.36532580760380206733382868882, 7.73102144096255023104365276788, 8.027105501047314182413259178583, 8.538687880420415041094117740367, 8.922941609669290152495843868069

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