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Properties

Label 4-325e2-1.1-c1e2-0-0
Degree $4$
Conductor $105625$
Sign $1$
Analytic cond. $6.73474$
Root an. cond. $1.61094$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 3·9-s − 6·13-s − 4·16-s − 4·17-s − 2·23-s + 14·27-s + 10·29-s + 12·39-s − 22·43-s + 8·48-s + 10·49-s + 8·51-s + 18·53-s + 14·61-s + 4·69-s − 10·79-s − 4·81-s − 20·87-s + 14·101-s − 2·103-s + 26·107-s − 2·113-s + 18·117-s + 22·121-s + 127-s + 44·129-s + ⋯
L(s)  = 1  − 1.15·3-s − 9-s − 1.66·13-s − 16-s − 0.970·17-s − 0.417·23-s + 2.69·27-s + 1.85·29-s + 1.92·39-s − 3.35·43-s + 1.15·48-s + 10/7·49-s + 1.12·51-s + 2.47·53-s + 1.79·61-s + 0.481·69-s − 1.12·79-s − 4/9·81-s − 2.14·87-s + 1.39·101-s − 0.197·103-s + 2.51·107-s − 0.188·113-s + 1.66·117-s + 2·121-s + 0.0887·127-s + 3.87·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(105625\)    =    \(5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(6.73474\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 105625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4384904774\)
\(L(\frac12)\) \(\approx\) \(0.4384904774\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad5 \( 1 \)
13$C_2$ \( 1 + 6 T + p T^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) 2.2.a_a
3$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.3.c_h
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.11.a_aw
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.17.e_bm
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.19.a_aw
23$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.23.c_bv
29$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.29.ak_df
31$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.31.a_bm
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.a_acs
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.a_s
43$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.43.w_hz
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.47.a_abe
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.53.as_hf
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.59.a_ade
61$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.61.ao_gp
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.67.a_k
71$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \) 2.71.a_abq
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.73.a_by
79$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.79.k_hb
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.83.a_afa
89$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.89.a_afm
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \) 2.97.a_ahi
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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

−11.93176266694298933058460370640, −11.51797952032722027855627646741, −11.08018920854494500666756217377, −10.36184021706980554193114109300, −10.19355003038935935112000501799, −9.703465973113253747598113452150, −8.867745395695592108036724025108, −8.442749595770913852285471177392, −8.426577148881573092317124951724, −7.20569607189996205962244713118, −7.02886080438866008384258629381, −6.47250326800870221202037188489, −5.96843779068964231574949881864, −5.37167074814700313175886168848, −4.76194707141289287292046691991, −4.68951004590417022965209666797, −3.56889414762037624882687025841, −2.61406413337654857103539023048, −2.27880256373408315256147343358, −0.48137645504867270155404486414, 0.48137645504867270155404486414, 2.27880256373408315256147343358, 2.61406413337654857103539023048, 3.56889414762037624882687025841, 4.68951004590417022965209666797, 4.76194707141289287292046691991, 5.37167074814700313175886168848, 5.96843779068964231574949881864, 6.47250326800870221202037188489, 7.02886080438866008384258629381, 7.20569607189996205962244713118, 8.426577148881573092317124951724, 8.442749595770913852285471177392, 8.867745395695592108036724025108, 9.703465973113253747598113452150, 10.19355003038935935112000501799, 10.36184021706980554193114109300, 11.08018920854494500666756217377, 11.51797952032722027855627646741, 11.93176266694298933058460370640

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