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Properties

Label 4-43904-1.1-c1e2-0-14
Degree $4$
Conductor $43904$
Sign $-1$
Analytic cond. $2.79935$
Root an. cond. $1.29349$
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 − 7-s − 8-s − 9-s + 14-s + 16-s − 3·17-s + 18-s − 15·23-s − 2·25-s − 28-s − 12·31-s − 32-s + 3·34-s − 36-s − 3·41-s + 15·46-s + 9·47-s + 49-s + 2·50-s + 56-s + 12·62-s + 63-s + 64-s − 3·68-s + 9·71-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 − 3.12·23-s − 2/5·25-s − 0.188·28-s − 2.15·31-s − 0.176·32-s + 0.514·34-s − 1/6·36-s − 0.468·41-s + 2.21·46-s + 1.31·47-s + 1/7·49-s + 0.282·50-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(43904\)    =    \(2^{7} \cdot 7^{3}\)
Sign: $-1$
Analytic conductor: \(2.79935\)
Root analytic conductor: \(1.29349\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 43904,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
7$C_1$ \( 1 + T \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.3.a_b
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.13.a_i
17$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.d_q
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.19.a_abi
23$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.23.p_dw
29$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \) 2.29.a_ar
31$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.m_dt
37$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.37.a_h
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.41.d_de
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.47.aj_cg
53$C_2^2$ \( 1 - 95 T^{2} + p^{2} T^{4} \) 2.53.a_adr
59$C_2^2$ \( 1 + 101 T^{2} + p^{2} T^{4} \) 2.59.a_dx
61$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \) 2.61.a_abo
67$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.67.a_bu
71$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.71.aj_ge
73$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.73.as_gw
79$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.79.af_fo
83$C_2^2$ \( 1 - 145 T^{2} + p^{2} T^{4} \) 2.83.a_afp
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.89.j_gw
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) 2.97.ay_mw
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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

−9.903259475993730301133601247932, −9.400946509192316215215642760736, −9.032872625967611552960835581299, −8.322439564034960678023719361421, −7.986082184770432412320687121706, −7.40607211260737083787304764859, −6.80999099230277597284059767890, −6.15879308148576935335469093848, −5.78902777848912703504487358241, −5.08639353379536435232603331513, −3.92316526135323766397833934784, −3.75383713546245393225954256150, −2.46401242958316858193289692726, −1.88856445010374489988253513893, 0, 1.88856445010374489988253513893, 2.46401242958316858193289692726, 3.75383713546245393225954256150, 3.92316526135323766397833934784, 5.08639353379536435232603331513, 5.78902777848912703504487358241, 6.15879308148576935335469093848, 6.80999099230277597284059767890, 7.40607211260737083787304764859, 7.986082184770432412320687121706, 8.322439564034960678023719361421, 9.032872625967611552960835581299, 9.400946509192316215215642760736, 9.903259475993730301133601247932

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