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Properties

Label 4-152488-1.1-c1e2-0-0
Degree $4$
Conductor $152488$
Sign $1$
Analytic cond. $9.72276$
Root an. cond. $1.76582$
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 + 4-s + 5·7-s − 8-s − 2·9-s − 3·11-s − 5·14-s + 16-s + 2·18-s + 3·22-s + 9·23-s − 4·25-s + 5·28-s + 6·29-s − 32-s − 2·36-s − 2·37-s − 8·43-s − 3·44-s − 9·46-s + 18·49-s + 4·50-s + 6·53-s − 5·56-s − 6·58-s − 10·63-s + 64-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 1.88·7-s − 0.353·8-s − 2/3·9-s − 0.904·11-s − 1.33·14-s + 1/4·16-s + 0.471·18-s + 0.639·22-s + 1.87·23-s − 4/5·25-s + 0.944·28-s + 1.11·29-s − 0.176·32-s − 1/3·36-s − 0.328·37-s − 1.21·43-s − 0.452·44-s − 1.32·46-s + 18/7·49-s + 0.565·50-s + 0.824·53-s − 0.668·56-s − 0.787·58-s − 1.25·63-s + 1/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152488 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152488 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(152488\)    =    \(2^{3} \cdot 7^{2} \cdot 389\)
Sign: $1$
Analytic conductor: \(9.72276\)
Root analytic conductor: \(1.76582\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 152488,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.337586839\)
\(L(\frac12)\) \(\approx\) \(1.337586839\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
389$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 15 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.5.a_e
11$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.d_e
13$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.13.a_au
17$C_2^2$ \( 1 - 23 T^{2} + p^{2} T^{4} \) 2.17.a_ax
19$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \) 2.19.a_bc
23$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.23.aj_cm
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.29.ag_cg
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.31.a_aba
37$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.37.c_bn
41$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.41.a_cg
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2^2$ \( 1 - 68 T^{2} + p^{2} T^{4} \) 2.47.a_acq
53$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.53.ag_db
59$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.59.a_au
61$C_2^2$ \( 1 + 91 T^{2} + p^{2} T^{4} \) 2.61.a_dn
67$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.67.at_io
71$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.ad_abm
73$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.73.a_af
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.79.ah_fu
83$C_2^2$ \( 1 - 23 T^{2} + p^{2} T^{4} \) 2.83.a_ax
89$C_2^2$ \( 1 + 106 T^{2} + p^{2} T^{4} \) 2.89.a_ec
97$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \) 2.97.a_t
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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

−9.186075654114003802297609429131, −8.577025373040178527533519726997, −8.295029626159765495561588837756, −8.098262088194789396778586674926, −7.40863619170219142319782333249, −7.04109579982492329874718514299, −6.43142294250561579503152927661, −5.62882662492342542752464257724, −5.16857453149356335678028132590, −4.93587968646227739191394426993, −4.12872920441806383286897367540, −3.22685752020164892318724473926, −2.53889129361503103527314251266, −1.89169188862927323439729918994, −0.908835149446819824255408357006, 0.908835149446819824255408357006, 1.89169188862927323439729918994, 2.53889129361503103527314251266, 3.22685752020164892318724473926, 4.12872920441806383286897367540, 4.93587968646227739191394426993, 5.16857453149356335678028132590, 5.62882662492342542752464257724, 6.43142294250561579503152927661, 7.04109579982492329874718514299, 7.40863619170219142319782333249, 8.098262088194789396778586674926, 8.295029626159765495561588837756, 8.577025373040178527533519726997, 9.186075654114003802297609429131

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