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Properties

Label 4-35712-1.1-c1e2-0-1
Degree $4$
Conductor $35712$
Sign $1$
Analytic cond. $2.27702$
Root an. cond. $1.22840$
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 + 7-s − 8-s + 9-s − 14-s + 16-s − 3·17-s − 18-s + 8·25-s + 28-s + 12·31-s − 32-s + 3·34-s + 36-s − 6·41-s + 15·47-s − 11·49-s − 8·50-s − 56-s − 12·62-s + 63-s + 64-s − 3·68-s + 9·71-s − 72-s + 13·73-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.267·14-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 8/5·25-s + 0.188·28-s + 2.15·31-s − 0.176·32-s + 0.514·34-s + 1/6·36-s − 0.937·41-s + 2.18·47-s − 1.57·49-s − 1.13·50-s − 0.133·56-s − 1.52·62-s + 0.125·63-s + 1/8·64-s − 0.363·68-s + 1.06·71-s − 0.117·72-s + 1.52·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35712 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35712 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(35712\)    =    \(2^{7} \cdot 3^{2} \cdot 31\)
Sign: $1$
Analytic conductor: \(2.27702\)
Root analytic conductor: \(1.22840\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 35712,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.001058052\)
\(L(\frac12)\) \(\approx\) \(1.001058052\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
31$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 11 T + p T^{2} ) \)
good5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.5.a_ai
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.ab_m
11$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.11.a_ai
13$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.13.a_ac
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.d_bi
19$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.19.a_n
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \) 2.29.a_n
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.a_aba
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.g_de
43$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \) 2.43.a_cm
47$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.47.ap_fs
53$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.53.a_b
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.a_aba
61$C_2^2$ \( 1 - 23 T^{2} + p^{2} T^{4} \) 2.61.a_ax
67$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.67.a_ack
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.71.aj_ec
73$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.73.an_he
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.c_da
83$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \) 2.83.a_cm
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.89.d_gw
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

−10.37643162125730736411343050895, −9.785450315068424579133430187506, −9.419857459359385796953068179790, −8.595878238414244091109677196943, −8.475597807256965775714814667096, −7.83977296215673643579225496329, −7.15191818760952813153619658034, −6.66033419143574310107310697447, −6.26836277750817205031839933208, −5.29603828646277411231281918514, −4.75949420709691509137038125127, −4.07127999702510228304528405887, −3.04607597007804813026438489080, −2.30459728818103735923905107578, −1.12071358707329301606513474448, 1.12071358707329301606513474448, 2.30459728818103735923905107578, 3.04607597007804813026438489080, 4.07127999702510228304528405887, 4.75949420709691509137038125127, 5.29603828646277411231281918514, 6.26836277750817205031839933208, 6.66033419143574310107310697447, 7.15191818760952813153619658034, 7.83977296215673643579225496329, 8.475597807256965775714814667096, 8.595878238414244091109677196943, 9.419857459359385796953068179790, 9.785450315068424579133430187506, 10.37643162125730736411343050895

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