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Properties

Label 4-623808-1.1-c1e2-0-12
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 6-s − 4·7-s + 8-s + 9-s − 12-s − 4·14-s + 16-s + 18-s + 2·19-s + 4·21-s − 24-s − 2·25-s − 27-s − 4·28-s − 6·29-s + 32-s + 36-s + 2·38-s + 14·41-s + 4·42-s + 4·43-s − 48-s − 2·49-s − 2·50-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.288·12-s − 1.06·14-s + 1/4·16-s + 0.235·18-s + 0.458·19-s + 0.872·21-s − 0.204·24-s − 2/5·25-s − 0.192·27-s − 0.755·28-s − 1.11·29-s + 0.176·32-s + 1/6·36-s + 0.324·38-s + 2.18·41-s + 0.617·42-s + 0.609·43-s − 0.144·48-s − 2/7·49-s − 0.282·50-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: \(0\)
Selberg data: \((4,\ 623808,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.642747061\)
\(L(\frac12)\) \(\approx\) \(1.642747061\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
3$C_1$ \( 1 + T \)
19$C_2$ \( 1 - 2 T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.7.e_s
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.17.a_w
23$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \) 2.23.a_abq
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.g_cg
31$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.31.a_ao
37$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.37.a_as
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.41.ao_es
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.43.ae_di
47$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.47.a_ak
53$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.s_gw
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.ai_cs
61$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.i_bm
67$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.67.a_abu
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) 2.71.am_fm
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.73.aq_hy
79$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.79.a_ac
83$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.83.a_gg
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) 2.89.ak_gw
97$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \) 2.97.a_du
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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

−8.145654901649601007440149498138, −7.79056387962926297798460496712, −7.47902692369469057041049508614, −6.74374945235889261602605085933, −6.53038867670044216467089513665, −6.13964380830864792272980375283, −5.63825240005367975557499299467, −5.26976181474469883458533862098, −4.62714010241034487789076615495, −4.09285294902583373683876654205, −3.56349389416153626283741913007, −3.14557192722245558817228296431, −2.48816393146487462114219431112, −1.71152780217248140295681386003, −0.58869815759023896820868809890, 0.58869815759023896820868809890, 1.71152780217248140295681386003, 2.48816393146487462114219431112, 3.14557192722245558817228296431, 3.56349389416153626283741913007, 4.09285294902583373683876654205, 4.62714010241034487789076615495, 5.26976181474469883458533862098, 5.63825240005367975557499299467, 6.13964380830864792272980375283, 6.53038867670044216467089513665, 6.74374945235889261602605085933, 7.47902692369469057041049508614, 7.79056387962926297798460496712, 8.145654901649601007440149498138

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