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Properties

Label 4-13600-1.1-c1e2-0-1
Degree $4$
Conductor $13600$
Sign $1$
Analytic cond. $0.867147$
Root an. cond. $0.964991$
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 − 8-s + 9-s + 7·13-s + 16-s − 17-s − 18-s − 5·25-s − 7·26-s − 12·29-s − 32-s + 34-s + 36-s + 22·37-s + 9·41-s − 4·49-s + 5·50-s + 7·52-s − 3·53-s + 12·58-s + 61-s + 64-s − 68-s − 72-s + 4·73-s − 22·74-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1/3·9-s + 1.94·13-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 25-s − 1.37·26-s − 2.22·29-s − 0.176·32-s + 0.171·34-s + 1/6·36-s + 3.61·37-s + 1.40·41-s − 4/7·49-s + 0.707·50-s + 0.970·52-s − 0.412·53-s + 1.57·58-s + 0.128·61-s + 1/8·64-s − 0.121·68-s − 0.117·72-s + 0.468·73-s − 2.55·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13600\)    =    \(2^{5} \cdot 5^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(0.867147\)
Root analytic conductor: \(0.964991\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 13600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8348881401\)
\(L(\frac12)\) \(\approx\) \(0.8348881401\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
5$C_2$ \( 1 + p T^{2} \)
17$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
good3$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.3.a_ab
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.7.a_e
11$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.11.a_af
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.13.ah_bk
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.23.a_w
29$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.29.m_dh
31$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.31.a_abm
37$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.37.aw_hn
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.41.aj_dw
43$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.43.a_aba
47$C_2^2$ \( 1 - 71 T^{2} + p^{2} T^{4} \) 2.47.a_act
53$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.d_dk
59$C_2^2$ \( 1 - 104 T^{2} + p^{2} T^{4} \) 2.59.a_aea
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.61.ab_eq
67$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.67.a_aba
71$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \) 2.71.a_fa
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.ae_g
79$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.79.a_bo
83$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.83.a_aby
89$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.89.a_abv
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

−11.14680265732505978918946867395, −10.99370414526814221038236976434, −9.919641626515640542128644787449, −9.567203220044579694443544543780, −9.120868614999567458547915133541, −8.423936815687109503508285129253, −7.80442618764508754500700424172, −7.52068820314411699379347478405, −6.50832086995624354061410190506, −6.05707593019747371132875710715, −5.54735130338473341869710101557, −4.21960741807223507739343400056, −3.80962316066387429759857659314, −2.60052844546027147490499082718, −1.38021899441878541523129607108, 1.38021899441878541523129607108, 2.60052844546027147490499082718, 3.80962316066387429759857659314, 4.21960741807223507739343400056, 5.54735130338473341869710101557, 6.05707593019747371132875710715, 6.50832086995624354061410190506, 7.52068820314411699379347478405, 7.80442618764508754500700424172, 8.423936815687109503508285129253, 9.120868614999567458547915133541, 9.567203220044579694443544543780, 9.919641626515640542128644787449, 10.99370414526814221038236976434, 11.14680265732505978918946867395

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