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Properties

Label 4-155e2-1.1-c1e2-0-6
Degree $4$
Conductor $24025$
Sign $-1$
Analytic cond. $1.53185$
Root an. cond. $1.11251$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 5·9-s − 5·11-s − 4·16-s − 4·25-s + 31-s + 3·41-s + 5·45-s + 5·49-s + 5·55-s + 5·59-s − 20·61-s − 71-s − 5·79-s + 4·80-s + 16·81-s + 20·89-s + 25·99-s + 4·101-s + 10·109-s − 121-s + 9·125-s + 127-s + 131-s + 137-s + 139-s + 20·144-s + ⋯
L(s)  = 1  − 0.447·5-s − 5/3·9-s − 1.50·11-s − 16-s − 4/5·25-s + 0.179·31-s + 0.468·41-s + 0.745·45-s + 5/7·49-s + 0.674·55-s + 0.650·59-s − 2.56·61-s − 0.118·71-s − 0.562·79-s + 0.447·80-s + 16/9·81-s + 2.11·89-s + 2.51·99-s + 0.398·101-s + 0.957·109-s − 0.0909·121-s + 0.804·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5/3·144-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(24025\)    =    \(5^{2} \cdot 31^{2}\)
Sign: $-1$
Analytic conductor: \(1.53185\)
Root analytic conductor: \(1.11251\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 24025,\ (\ :1/2, 1/2),\ -1)\)

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 $\mathbf{F}_p$
bad5$C_2$ \( 1 + T + p T^{2} \)
31$C_2$ \( 1 - T + p T^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) 2.2.a_a
3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.3.a_f
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.7.a_af
11$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.f_ba
13$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \) 2.13.a_p
17$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \) 2.17.a_p
19$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.19.a_n
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.29.a_bh
37$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \) 2.37.a_p
41$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.ad_bc
43$C_2^2$ \( 1 - 55 T^{2} + p^{2} T^{4} \) 2.43.a_acd
47$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.47.a_ak
53$C_2^2$ \( 1 + 95 T^{2} + p^{2} T^{4} \) 2.53.a_dr
59$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) 2.59.af_eo
61$C_2$$\times$$C_2$ \( ( 1 + 9 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.61.u_in
67$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.67.a_af
71$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.71.b_fa
73$C_2^2$ \( 1 + 95 T^{2} + p^{2} T^{4} \) 2.73.a_dr
79$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.79.f_gc
83$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.83.a_f
89$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.89.au_jt
97$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.97.a_by
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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

−10.49531050357516260305943060702, −10.04932007407217166263027352722, −9.158437888058015058882604823736, −8.883398855474776663154500905999, −8.261039098680368079704847747004, −7.70591290018467291375900181263, −7.39686422093544926567567527641, −6.35230861097195661660111035305, −5.94157911424031664187917838550, −5.22860448473015629426720465022, −4.71227256176263371685409066144, −3.75664551448942402655286428047, −2.87877559651773673301788270026, −2.31956228278278751748382753816, 0, 2.31956228278278751748382753816, 2.87877559651773673301788270026, 3.75664551448942402655286428047, 4.71227256176263371685409066144, 5.22860448473015629426720465022, 5.94157911424031664187917838550, 6.35230861097195661660111035305, 7.39686422093544926567567527641, 7.70591290018467291375900181263, 8.261039098680368079704847747004, 8.883398855474776663154500905999, 9.158437888058015058882604823736, 10.04932007407217166263027352722, 10.49531050357516260305943060702

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