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Properties

Label 4-623808-1.1-c1e2-0-40
Degree $4$
Conductor $623808$
Sign $-1$
Analytic cond. $39.7745$
Root an. cond. $2.51131$
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  + 3-s − 6·7-s + 9-s − 4·13-s + 2·19-s − 6·21-s − 25-s + 27-s − 4·31-s + 16·37-s − 4·39-s + 26·43-s + 13·49-s + 2·57-s − 26·61-s − 6·63-s + 8·67-s − 6·73-s − 75-s − 8·79-s + 81-s + 24·91-s − 4·93-s + 4·97-s − 12·103-s + 32·109-s + 16·111-s + ⋯
L(s)  = 1  + 0.577·3-s − 2.26·7-s + 1/3·9-s − 1.10·13-s + 0.458·19-s − 1.30·21-s − 1/5·25-s + 0.192·27-s − 0.718·31-s + 2.63·37-s − 0.640·39-s + 3.96·43-s + 13/7·49-s + 0.264·57-s − 3.32·61-s − 0.755·63-s + 0.977·67-s − 0.702·73-s − 0.115·75-s − 0.900·79-s + 1/9·81-s + 2.51·91-s − 0.414·93-s + 0.406·97-s − 1.18·103-s + 3.06·109-s + 1.51·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(623808\)    =    \(2^{6} \cdot 3^{3} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(39.7745\)
Root analytic conductor: \(2.51131\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 623808,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3$C_1$ \( 1 - T \)
19$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
7$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.7.g_x
11$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.11.a_v
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
17$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.17.a_j
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.31.e_co
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.37.aq_fi
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.a_s
43$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \) 2.43.aba_jv
47$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.47.a_acx
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.61.ba_lf
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.a_da
73$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.73.g_fz
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.79.i_gs
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.a_fu
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−7.924107656862495380801900682255, −7.74468446164444943887517800732, −7.21252020459675544867285019060, −7.00612622598426407511060309816, −6.16799098038826968846498615960, −6.02515524155855065259322751403, −5.66548080355213698116568009078, −4.59164546840578345262367580791, −4.44688717155938259742687854966, −3.65611432799862238242056225439, −3.25222241798197103354320694490, −2.53070795011250446615850135913, −2.49847213158180922818846804858, −1.07996559647171411835733236564, 0, 1.07996559647171411835733236564, 2.49847213158180922818846804858, 2.53070795011250446615850135913, 3.25222241798197103354320694490, 3.65611432799862238242056225439, 4.44688717155938259742687854966, 4.59164546840578345262367580791, 5.66548080355213698116568009078, 6.02515524155855065259322751403, 6.16799098038826968846498615960, 7.00612622598426407511060309816, 7.21252020459675544867285019060, 7.74468446164444943887517800732, 7.924107656862495380801900682255

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