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Properties

Label 4-513e2-1.1-c1e2-0-6
Degree $4$
Conductor $263169$
Sign $1$
Analytic cond. $16.7798$
Root an. cond. $2.02393$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2·4-s + 2·5-s − 3·7-s − 5·8-s − 2·10-s + 3·11-s + 6·13-s + 3·14-s + 5·16-s + 3·17-s − 8·19-s + 4·20-s − 3·22-s + 8·23-s − 7·25-s − 6·26-s − 6·28-s + 10·29-s + 7·31-s − 10·32-s − 3·34-s − 6·35-s + 4·37-s + 8·38-s − 10·40-s + 2·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 4-s + 0.894·5-s − 1.13·7-s − 1.76·8-s − 0.632·10-s + 0.904·11-s + 1.66·13-s + 0.801·14-s + 5/4·16-s + 0.727·17-s − 1.83·19-s + 0.894·20-s − 0.639·22-s + 1.66·23-s − 7/5·25-s − 1.17·26-s − 1.13·28-s + 1.85·29-s + 1.25·31-s − 1.76·32-s − 0.514·34-s − 1.01·35-s + 0.657·37-s + 1.29·38-s − 1.58·40-s + 0.312·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 263169 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 263169 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(263169\)    =    \(3^{6} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(16.7798\)
Root analytic conductor: \(2.02393\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 263169,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.489056452\)
\(L(\frac12)\) \(\approx\) \(1.489056452\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_2$ \( 1 + 8 T + p T^{2} \)
good2$C_2^2$ \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) 2.2.b_ab
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.5.ac_l
7$C_2^2$ \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.7.d_c
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.11.ad_ac
13$C_2^2$ \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.13.ag_x
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.17.ad_ai
23$C_2^2$ \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.23.ai_bp
29$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.29.ak_df
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.ah_s
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.41.ac_df
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.43.i_v
47$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.47.s_gt
53$C_2^2$ \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.53.ad_abs
59$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.59.g_ex
61$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.61.ao_gp
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_abz
71$C_2^2$ \( 1 + 15 T + 154 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.71.p_fy
73$C_2^2$ \( 1 - 5 T - 48 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.73.af_abw
79$C_2^2$ \( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.79.am_cn
83$C_2^2$ \( 1 + T - 82 T^{2} + p T^{3} + p^{2} T^{4} \) 2.83.b_ade
89$C_2^2$ \( 1 + T - 88 T^{2} + p T^{3} + p^{2} T^{4} \) 2.89.b_adk
97$C_2^2$ \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.97.ac_adp
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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

−11.22764068267892271213048943580, −10.53982779185925441555393422261, −9.977551879038831302476544619924, −9.914733318800735856680820690356, −9.422274484681160721014995885668, −8.849058922344088454146548866671, −8.443471412362247655743573755376, −8.396419621598843879061991890148, −7.45918675433093412309689321841, −6.59846699055009858271107852262, −6.58985942488832300076487238506, −6.13226787349324150529786119100, −6.06570041081805960450976585646, −5.17165467997458954996451952542, −4.33990420847873951360144415563, −3.47959918976036542031719361535, −3.24570181529285020852700786694, −2.50510013608551059842848175087, −1.75576313996937568804761698360, −0.837857107130942695788145712451, 0.837857107130942695788145712451, 1.75576313996937568804761698360, 2.50510013608551059842848175087, 3.24570181529285020852700786694, 3.47959918976036542031719361535, 4.33990420847873951360144415563, 5.17165467997458954996451952542, 6.06570041081805960450976585646, 6.13226787349324150529786119100, 6.58985942488832300076487238506, 6.59846699055009858271107852262, 7.45918675433093412309689321841, 8.396419621598843879061991890148, 8.443471412362247655743573755376, 8.849058922344088454146548866671, 9.422274484681160721014995885668, 9.914733318800735856680820690356, 9.977551879038831302476544619924, 10.53982779185925441555393422261, 11.22764068267892271213048943580

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