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Properties

Label 4-405e2-1.1-c1e2-0-14
Degree $4$
Conductor $164025$
Sign $1$
Analytic cond. $10.4583$
Root an. cond. $1.79831$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·4-s − 6·7-s − 4·13-s + 5·16-s − 16·19-s + 25-s + 18·28-s − 8·37-s − 16·43-s + 13·49-s + 12·52-s + 14·61-s − 3·64-s − 6·67-s + 8·73-s + 48·76-s − 12·79-s + 24·91-s + 4·97-s − 3·100-s + 16·103-s + 10·109-s − 30·112-s − 18·121-s + 127-s + 131-s + 96·133-s + ⋯
L(s)  = 1  − 3/2·4-s − 2.26·7-s − 1.10·13-s + 5/4·16-s − 3.67·19-s + 1/5·25-s + 3.40·28-s − 1.31·37-s − 2.43·43-s + 13/7·49-s + 1.66·52-s + 1.79·61-s − 3/8·64-s − 0.733·67-s + 0.936·73-s + 5.50·76-s − 1.35·79-s + 2.51·91-s + 0.406·97-s − 0.299·100-s + 1.57·103-s + 0.957·109-s − 2.83·112-s − 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 8.32·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(164025\)    =    \(3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(10.4583\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 164025,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.2.a_d
7$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.7.g_x
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.a_s
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
19$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.19.q_dy
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.23.a_bl
29$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.29.a_cf
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.37.i_dm
41$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.41.a_cf
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.43.q_fu
47$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.47.a_bt
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.53.a_dy
59$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.59.a_ada
61$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.61.ao_gp
67$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.67.g_fn
71$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.71.a_fi
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.73.ai_gg
79$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.79.m_hm
83$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.83.a_dh
89$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.89.a_abv
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−8.964305577132556078601253128051, −8.406907006225460062267410041104, −8.053933674867411547003077582597, −6.95907981187563203473208917697, −6.84595624283791199980948871072, −6.29557968587596560657178728455, −5.86971236166707955375003452449, −4.84628635731234228377894729907, −4.82687436247101660115281422977, −3.77402379572268668385012510927, −3.75836640301739809806939141128, −2.80534791056945823123328757931, −2.06014522909482997436709887093, 0, 0, 2.06014522909482997436709887093, 2.80534791056945823123328757931, 3.75836640301739809806939141128, 3.77402379572268668385012510927, 4.82687436247101660115281422977, 4.84628635731234228377894729907, 5.86971236166707955375003452449, 6.29557968587596560657178728455, 6.84595624283791199980948871072, 6.95907981187563203473208917697, 8.053933674867411547003077582597, 8.406907006225460062267410041104, 8.964305577132556078601253128051

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