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Properties

Label 4-630e2-1.1-c1e2-0-31
Degree $4$
Conductor $396900$
Sign $1$
Analytic cond. $25.3066$
Root an. cond. $2.24289$
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  + 2·2-s + 3·4-s − 4·7-s + 4·8-s − 8·14-s + 5·16-s + 25-s − 12·28-s + 12·29-s + 6·32-s + 4·37-s − 8·43-s + 9·49-s + 2·50-s + 12·53-s − 16·56-s + 24·58-s + 7·64-s − 8·67-s + 8·74-s + 16·79-s − 16·86-s + 18·98-s + 3·100-s + 24·106-s + 24·107-s − 20·109-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 1.51·7-s + 1.41·8-s − 2.13·14-s + 5/4·16-s + 1/5·25-s − 2.26·28-s + 2.22·29-s + 1.06·32-s + 0.657·37-s − 1.21·43-s + 9/7·49-s + 0.282·50-s + 1.64·53-s − 2.13·56-s + 3.15·58-s + 7/8·64-s − 0.977·67-s + 0.929·74-s + 1.80·79-s − 1.72·86-s + 1.81·98-s + 3/10·100-s + 2.33·106-s + 2.32·107-s − 1.91·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(396900\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(25.3066\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 396900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.913565526\)
\(L(\frac12)\) \(\approx\) \(3.913565526\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.a_ac
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.a_fm
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.89.a_afq
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.a_hi
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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

−8.644229019078172602684577487278, −8.145462184294923465930650647398, −7.51828953604385192982905489512, −6.97913667513999229030385049043, −6.73106470746564203527567325446, −6.11397905422548756742635703073, −6.03795573121967783311486434379, −5.26570153655431196939565550355, −4.75701556135232590446975582633, −4.32687879913691587957737883725, −3.63359445531324431774910445192, −3.21826656477469471981988966496, −2.74628747875934588873509462774, −2.10133345305515366205304847130, −0.879817280069523657260066044025, 0.879817280069523657260066044025, 2.10133345305515366205304847130, 2.74628747875934588873509462774, 3.21826656477469471981988966496, 3.63359445531324431774910445192, 4.32687879913691587957737883725, 4.75701556135232590446975582633, 5.26570153655431196939565550355, 6.03795573121967783311486434379, 6.11397905422548756742635703073, 6.73106470746564203527567325446, 6.97913667513999229030385049043, 7.51828953604385192982905489512, 8.145462184294923465930650647398, 8.644229019078172602684577487278

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