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Properties

Label 4-623808-1.1-c1e2-0-50
Degree $4$
Conductor $623808$
Sign $-1$
Analytic cond. $39.7745$
Root an. cond. $2.51131$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·7-s + 9-s + 2·13-s + 6·19-s + 2·21-s + 2·25-s − 27-s + 4·31-s − 8·37-s − 2·39-s − 12·43-s − 10·49-s − 6·57-s − 20·61-s − 2·63-s + 10·67-s + 12·73-s − 2·75-s − 18·79-s + 81-s − 4·91-s − 4·93-s − 8·97-s + 2·103-s − 10·109-s + 8·111-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.554·13-s + 1.37·19-s + 0.436·21-s + 2/5·25-s − 0.192·27-s + 0.718·31-s − 1.31·37-s − 0.320·39-s − 1.82·43-s − 1.42·49-s − 0.794·57-s − 2.56·61-s − 0.251·63-s + 1.22·67-s + 1.40·73-s − 0.230·75-s − 2.02·79-s + 1/9·81-s − 0.419·91-s − 0.414·93-s − 0.812·97-s + 0.197·103-s − 0.957·109-s + 0.759·111-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: yes
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$C_2$ \( 1 - 6 T + p T^{2} \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.c_o
11$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.11.a_e
13$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.ac_s
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
23$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.23.a_ai
29$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.29.a_bi
31$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.31.ae_ck
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.i_cc
41$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.41.a_w
43$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.m_eo
47$C_2^2$ \( 1 - 64 T^{2} + p^{2} T^{4} \) 2.47.a_acm
53$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.53.a_bq
59$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.59.a_aw
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.67.ak_eg
71$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.71.a_g
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.73.am_gk
79$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.79.s_ig
83$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.83.a_u
89$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \) 2.89.a_adm
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.i_gs
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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

−8.108968848734086284000726693486, −7.78499932365386369687261683110, −7.01844215709262667167203378115, −6.82956747584740142949100046518, −6.41897478231044336642206137228, −5.86027847453465385804328226204, −5.44508730156572028295813416190, −4.91240861006383619654100874468, −4.53134186539358396795441458009, −3.70758402662339169747033540804, −3.26893534629780376248160237436, −2.89839453673235949497987425818, −1.80666670928817171290268325841, −1.16736969930187908758058462808, 0, 1.16736969930187908758058462808, 1.80666670928817171290268325841, 2.89839453673235949497987425818, 3.26893534629780376248160237436, 3.70758402662339169747033540804, 4.53134186539358396795441458009, 4.91240861006383619654100874468, 5.44508730156572028295813416190, 5.86027847453465385804328226204, 6.41897478231044336642206137228, 6.82956747584740142949100046518, 7.01844215709262667167203378115, 7.78499932365386369687261683110, 8.108968848734086284000726693486

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