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Properties

Label 4-171e2-1.1-c1e2-0-3
Degree $4$
Conductor $29241$
Sign $1$
Analytic cond. $1.86443$
Root an. cond. $1.16852$
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  − 4-s + 2·7-s + 6·13-s − 3·16-s − 2·19-s + 4·25-s − 2·28-s + 6·31-s + 4·43-s − 2·49-s − 6·52-s + 8·61-s + 7·64-s − 6·67-s + 4·73-s + 2·76-s − 12·79-s + 12·91-s + 18·97-s − 4·100-s − 24·103-s + 18·109-s − 6·112-s − 14·121-s − 6·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.755·7-s + 1.66·13-s − 3/4·16-s − 0.458·19-s + 4/5·25-s − 0.377·28-s + 1.07·31-s + 0.609·43-s − 2/7·49-s − 0.832·52-s + 1.02·61-s + 7/8·64-s − 0.733·67-s + 0.468·73-s + 0.229·76-s − 1.35·79-s + 1.25·91-s + 1.82·97-s − 2/5·100-s − 2.36·103-s + 1.72·109-s − 0.566·112-s − 1.27·121-s − 0.538·124-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29241 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29241 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(29241\)    =    \(3^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1.86443\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 29241,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.244872874\)
\(L(\frac12)\) \(\approx\) \(1.244872874\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 2 T + p T^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.2.a_b
5$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.5.a_ae
7$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.ac_g
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
13$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.13.ag_bi
17$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.17.a_u
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.23.a_c
29$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.29.a_abm
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.31.ag_bu
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.41.a_w
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.ae_cc
47$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.47.a_acs
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.a_abm
59$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.59.a_w
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.ai_dy
67$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.g_fm
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.m_hi
83$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.83.a_di
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.97.as_is
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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

−10.67158236312907942011855357669, −10.06619629051700937204504733891, −9.364494833175988349524653486455, −8.854691926635995753535836530792, −8.454130566403482124577959481375, −8.099565268475924332754168128682, −7.31027405473375894911311979415, −6.62191940488887730442440752752, −6.15725082783879973463225430250, −5.42396522637242878317650225643, −4.70130152684274507459740347207, −4.23624815350723641098158986501, −3.47153083484185323981713952561, −2.43017794474286740532759664842, −1.23659485968081848521151304536, 1.23659485968081848521151304536, 2.43017794474286740532759664842, 3.47153083484185323981713952561, 4.23624815350723641098158986501, 4.70130152684274507459740347207, 5.42396522637242878317650225643, 6.15725082783879973463225430250, 6.62191940488887730442440752752, 7.31027405473375894911311979415, 8.099565268475924332754168128682, 8.454130566403482124577959481375, 8.854691926635995753535836530792, 9.364494833175988349524653486455, 10.06619629051700937204504733891, 10.67158236312907942011855357669

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