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Properties

Label 4-194688-1.1-c1e2-0-12
Degree $4$
Conductor $194688$
Sign $1$
Analytic cond. $12.4134$
Root an. cond. $1.87703$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s + 9-s + 2·13-s + 4·17-s + 2·25-s + 12·29-s − 4·37-s + 12·41-s + 4·45-s − 14·49-s + 12·53-s − 4·61-s + 8·65-s − 28·73-s + 81-s + 16·85-s − 36·89-s − 12·97-s + 28·101-s − 4·109-s − 12·113-s + 2·117-s − 22·121-s − 28·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 1.78·5-s + 1/3·9-s + 0.554·13-s + 0.970·17-s + 2/5·25-s + 2.22·29-s − 0.657·37-s + 1.87·41-s + 0.596·45-s − 2·49-s + 1.64·53-s − 0.512·61-s + 0.992·65-s − 3.27·73-s + 1/9·81-s + 1.73·85-s − 3.81·89-s − 1.21·97-s + 2.78·101-s − 0.383·109-s − 1.12·113-s + 0.184·117-s − 2·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 194688 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194688 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(194688\)    =    \(2^{7} \cdot 3^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(12.4134\)
Root analytic conductor: \(1.87703\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 194688,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.664572713\)
\(L(\frac12)\) \(\approx\) \(2.664572713\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.5.ae_o
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.17.ae_bm
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.43.a_acg
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.47.a_da
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.a_cc
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.73.bc_ne
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.83.a_dy
89$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \) 2.89.bk_ti
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.97.m_iw
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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

−9.067897300263890057771705983710, −8.860624431335414724430887662727, −8.023927242894012661962773978846, −7.905269575500993133277771362341, −6.92038776336112574405112626570, −6.75712783178321908907460309814, −5.93233764524938909111793115936, −5.82101553029460348251109404035, −5.35188862470330772210371090221, −4.52332315712058664242014926650, −4.11794986731058490261504789900, −3.09863617379496380583398157204, −2.68084777327604974871877197850, −1.76107356520720300622704659457, −1.20573633415257204749821945410, 1.20573633415257204749821945410, 1.76107356520720300622704659457, 2.68084777327604974871877197850, 3.09863617379496380583398157204, 4.11794986731058490261504789900, 4.52332315712058664242014926650, 5.35188862470330772210371090221, 5.82101553029460348251109404035, 5.93233764524938909111793115936, 6.75712783178321908907460309814, 6.92038776336112574405112626570, 7.905269575500993133277771362341, 8.023927242894012661962773978846, 8.860624431335414724430887662727, 9.067897300263890057771705983710

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