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Properties

Label 4-38464-1.1-c1e2-0-1
Degree $4$
Conductor $38464$
Sign $-1$
Analytic cond. $2.45249$
Root an. cond. $1.25141$
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  − 2-s − 4-s − 3·7-s + 3·8-s − 9-s + 3·14-s − 16-s − 3·17-s + 18-s + 6·23-s + 3·25-s + 3·28-s − 31-s − 5·32-s + 3·34-s + 36-s − 16·41-s − 6·46-s − 8·47-s − 5·49-s − 3·50-s − 9·56-s + 62-s + 3·63-s + 7·64-s + 3·68-s + 4·71-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 1.13·7-s + 1.06·8-s − 1/3·9-s + 0.801·14-s − 1/4·16-s − 0.727·17-s + 0.235·18-s + 1.25·23-s + 3/5·25-s + 0.566·28-s − 0.179·31-s − 0.883·32-s + 0.514·34-s + 1/6·36-s − 2.49·41-s − 0.884·46-s − 1.16·47-s − 5/7·49-s − 0.424·50-s − 1.20·56-s + 0.127·62-s + 0.377·63-s + 7/8·64-s + 0.363·68-s + 0.474·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(38464\)    =    \(2^{6} \cdot 601\)
Sign: $-1$
Analytic conductor: \(2.45249\)
Root analytic conductor: \(1.25141\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 38464,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T + p T^{2} \)
601$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 23 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.3.a_b
5$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.5.a_ad
7$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.d_o
11$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.11.a_aq
13$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.13.a_f
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.d_bi
19$C_2^2$ \( 1 - 24 T^{2} + p^{2} T^{4} \) 2.19.a_ay
23$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.23.ag_t
29$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \) 2.29.a_abs
31$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.31.b_ci
37$C_2^2$ \( 1 + 41 T^{2} + p^{2} T^{4} \) 2.37.a_bp
41$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.q_fm
43$C_2^2$ \( 1 - 36 T^{2} + p^{2} T^{4} \) 2.43.a_abk
47$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.i_ec
53$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \) 2.53.a_bs
59$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.59.a_u
61$C_2^2$ \( 1 + 67 T^{2} + p^{2} T^{4} \) 2.61.a_cp
67$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.67.a_bm
71$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.71.ae_fm
73$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.w_jy
79$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.79.u_io
83$C_2^2$ \( 1 - 96 T^{2} + p^{2} T^{4} \) 2.83.a_ads
89$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.89.ag_br
97$C_2$$\times$$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.97.x_mm
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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

−9.978628431584290057021336499547, −9.586201167676856312602363585276, −8.911453914767930074351399060074, −8.635129987664542132560276327759, −8.233456335650843499811957589176, −7.27521436980339770569471937733, −6.98200449387702706154369605062, −6.41026384897749176970242258782, −5.65492036219451761385841707146, −4.92809104564400681130806179584, −4.44868416323893516951229556129, −3.42580354915875859161626738046, −2.96119619316935944052450790664, −1.59296124216214222571315302611, 0, 1.59296124216214222571315302611, 2.96119619316935944052450790664, 3.42580354915875859161626738046, 4.44868416323893516951229556129, 4.92809104564400681130806179584, 5.65492036219451761385841707146, 6.41026384897749176970242258782, 6.98200449387702706154369605062, 7.27521436980339770569471937733, 8.233456335650843499811957589176, 8.635129987664542132560276327759, 8.911453914767930074351399060074, 9.586201167676856312602363585276, 9.978628431584290057021336499547

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