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Properties

Label 4-16256-1.1-c1e2-0-1
Degree $4$
Conductor $16256$
Sign $1$
Analytic cond. $1.03649$
Root an. cond. $1.00900$
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 − 2·7-s + 8-s + 4·9-s − 2·14-s + 16-s + 4·18-s − 6·23-s + 8·25-s − 2·28-s − 8·31-s + 32-s + 4·36-s − 12·41-s − 6·46-s + 12·47-s − 2·49-s + 8·50-s − 2·56-s − 8·62-s − 8·63-s + 64-s + 12·71-s + 4·72-s − 8·73-s − 8·79-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s + 4/3·9-s − 0.534·14-s + 1/4·16-s + 0.942·18-s − 1.25·23-s + 8/5·25-s − 0.377·28-s − 1.43·31-s + 0.176·32-s + 2/3·36-s − 1.87·41-s − 0.884·46-s + 1.75·47-s − 2/7·49-s + 1.13·50-s − 0.267·56-s − 1.01·62-s − 1.00·63-s + 1/8·64-s + 1.42·71-s + 0.471·72-s − 0.936·73-s − 0.900·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16256\)    =    \(2^{7} \cdot 127\)
Sign: $1$
Analytic conductor: \(1.03649\)
Root analytic conductor: \(1.00900\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 16256,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.652905884\)
\(L(\frac12)\) \(\approx\) \(1.652905884\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
127$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 16 T + p T^{2} ) \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.3.a_ae
5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.5.a_ai
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.c_g
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.a_ao
13$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.13.a_ac
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.g_bu
29$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \) 2.29.a_abg
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.37.a_ao
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.41.m_de
43$C_2^2$ \( 1 + 76 T^{2} + p^{2} T^{4} \) 2.43.a_cy
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) 2.47.am_dq
53$C_2^2$ \( 1 + 52 T^{2} + p^{2} T^{4} \) 2.53.a_ca
59$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \) 2.59.a_adc
61$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.61.a_aba
67$C_2^2$ \( 1 + 112 T^{2} + p^{2} T^{4} \) 2.67.a_ei
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.71.am_gw
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.i_ew
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.79.i_be
83$C_2^2$ \( 1 + 76 T^{2} + p^{2} T^{4} \) 2.83.a_cy
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.89.s_gw
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_li
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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

−10.96743957298042424492697871954, −10.60794985209562441519864745421, −9.916963526334587548850855441002, −9.708058378789662944481742506607, −8.852694537960833244680256860450, −8.298321819382308291768630668734, −7.39759956705511640157286921047, −7.02247494634365626714113273141, −6.55490725742751218720735463028, −5.76287066970347225618581553832, −5.12521620280023890407290336156, −4.28305941353259536785090560605, −3.76859983368908244233108651348, −2.87953762949568491699482050895, −1.68940400522491619144577409317, 1.68940400522491619144577409317, 2.87953762949568491699482050895, 3.76859983368908244233108651348, 4.28305941353259536785090560605, 5.12521620280023890407290336156, 5.76287066970347225618581553832, 6.55490725742751218720735463028, 7.02247494634365626714113273141, 7.39759956705511640157286921047, 8.298321819382308291768630668734, 8.852694537960833244680256860450, 9.708058378789662944481742506607, 9.916963526334587548850855441002, 10.60794985209562441519864745421, 10.96743957298042424492697871954

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