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Properties

Label 4-373527-1.1-c1e2-0-9
Degree $4$
Conductor $373527$
Sign $-1$
Analytic cond. $23.8164$
Root an. cond. $2.20911$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 4·4-s + 2·5-s − 7-s + 6·9-s − 12·12-s + 6·15-s + 12·16-s − 4·17-s − 8·20-s − 3·21-s − 7·25-s + 9·27-s + 4·28-s − 2·35-s − 24·36-s − 10·37-s + 4·41-s − 16·43-s + 12·45-s − 16·47-s + 36·48-s + 49-s − 12·51-s − 6·59-s − 24·60-s − 6·63-s + ⋯
L(s)  = 1  + 1.73·3-s − 2·4-s + 0.894·5-s − 0.377·7-s + 2·9-s − 3.46·12-s + 1.54·15-s + 3·16-s − 0.970·17-s − 1.78·20-s − 0.654·21-s − 7/5·25-s + 1.73·27-s + 0.755·28-s − 0.338·35-s − 4·36-s − 1.64·37-s + 0.624·41-s − 2.43·43-s + 1.78·45-s − 2.33·47-s + 5.19·48-s + 1/7·49-s − 1.68·51-s − 0.781·59-s − 3.09·60-s − 0.755·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373527 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373527 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(373527\)    =    \(3^{2} \cdot 7^{3} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(23.8164\)
Root analytic conductor: \(2.20911\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 373527,\ (\ :1/2, 1/2),\ -1)\)

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 $\mathbf{F}_p$
bad3$C_2$ \( 1 - p T + p T^{2} \)
7$C_1$ \( 1 + T \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.2.a_e
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.5.ac_l
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.17.e_bm
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.19.a_c
23$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.23.a_v
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.a_abq
31$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.31.a_cj
37$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.37.k_dv
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.43.q_fu
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.47.q_gc
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.59.g_ex
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.a_eo
67$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.67.g_fn
71$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.71.a_fl
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.a_bu
79$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.79.am_hm
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.83.y_ly
89$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \) 2.89.abe_pn
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.97.a_gn
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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

−8.513651119993219671602342798866, −8.333198695137771254103563674887, −7.66866498238269472650919646245, −7.41775134048211629573821110448, −6.42370236549041572726553632076, −6.27653205655726142636171995192, −5.38913047752314397636287568574, −4.95399628290113050953278162390, −4.53418395180193611114078817717, −3.85266175099960933742282200478, −3.48522622489004363314719629746, −3.04533793513601036750822233008, −2.00735365908684561839702035924, −1.58751343454368774869669820755, 0, 1.58751343454368774869669820755, 2.00735365908684561839702035924, 3.04533793513601036750822233008, 3.48522622489004363314719629746, 3.85266175099960933742282200478, 4.53418395180193611114078817717, 4.95399628290113050953278162390, 5.38913047752314397636287568574, 6.27653205655726142636171995192, 6.42370236549041572726553632076, 7.41775134048211629573821110448, 7.66866498238269472650919646245, 8.333198695137771254103563674887, 8.513651119993219671602342798866

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