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Properties

Label 4-155e2-1.1-c1e2-0-3
Degree $4$
Conductor $24025$
Sign $1$
Analytic cond. $1.53185$
Root an. cond. $1.11251$
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  + 4·2-s − 2·3-s + 8·4-s + 5-s − 8·6-s + 2·7-s + 8·8-s + 3·9-s + 4·10-s − 11-s − 16·12-s + 8·14-s − 2·15-s − 4·16-s + 2·17-s + 12·18-s + 8·19-s + 8·20-s − 4·21-s − 4·22-s − 8·23-s − 16·24-s − 10·27-s + 16·28-s − 6·29-s − 8·30-s + 11·31-s + ⋯
L(s)  = 1  + 2.82·2-s − 1.15·3-s + 4·4-s + 0.447·5-s − 3.26·6-s + 0.755·7-s + 2.82·8-s + 9-s + 1.26·10-s − 0.301·11-s − 4.61·12-s + 2.13·14-s − 0.516·15-s − 16-s + 0.485·17-s + 2.82·18-s + 1.83·19-s + 1.78·20-s − 0.872·21-s − 0.852·22-s − 1.66·23-s − 3.26·24-s − 1.92·27-s + 3.02·28-s − 1.11·29-s − 1.46·30-s + 1.97·31-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: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 24025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.564611140\)
\(L(\frac12)\) \(\approx\) \(3.564611140\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
31$C_2$ \( 1 - 11 T + p T^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )^{2} \) 2.2.ae_i
3$C_2^2$ \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_b
7$C_2^2$ \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_ad
11$C_2^2$ \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) 2.11.b_ak
13$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.13.a_an
17$C_2^2$ \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.17.ac_an
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.19.ai_bt
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.23.i_ck
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.29.g_cp
37$C_2^2$ \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.37.i_bb
41$C_2^2$ \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.41.ad_abg
43$C_2^2$ \( 1 + 6 T - 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.43.g_ah
47$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.47.ae_du
53$C_2^2$ \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.53.am_dn
59$C_2^2$ \( 1 + 5 T - 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.59.f_abi
61$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.61.ac_et
67$C_2^2$ \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_ad
71$C_2^2$ \( 1 + 15 T + 154 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.71.p_fy
73$C_2^2$ \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_acr
79$C_2^2$ \( 1 - 9 T + 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.79.aj_c
83$C_2^2$ \( 1 - 4 T - 67 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.83.ae_acp
89$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.89.o_it
97$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.97.y_na
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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

−13.55593111315306706842237051223, −12.65621841923551100758798437345, −12.19563478936745304193479045732, −11.96423064813585483616133106085, −11.56655391900895207109645015501, −11.25944302888091688369632982575, −10.38973904995505962643804707823, −9.945597895644451813523708626220, −9.303836363939542982619309428737, −8.437030060057162387642714551579, −7.47087177994057837339723874502, −7.15389918782590395811539451669, −6.16627312211400741566289817140, −5.87016029516199940113803405452, −5.43789700431921609813405892794, −5.07217617723638916400212916653, −4.35985557642864298962322164043, −3.85390365218689848036517225980, −3.03108790941852639483158102348, −1.94805695369951299108053156215, 1.94805695369951299108053156215, 3.03108790941852639483158102348, 3.85390365218689848036517225980, 4.35985557642864298962322164043, 5.07217617723638916400212916653, 5.43789700431921609813405892794, 5.87016029516199940113803405452, 6.16627312211400741566289817140, 7.15389918782590395811539451669, 7.47087177994057837339723874502, 8.437030060057162387642714551579, 9.303836363939542982619309428737, 9.945597895644451813523708626220, 10.38973904995505962643804707823, 11.25944302888091688369632982575, 11.56655391900895207109645015501, 11.96423064813585483616133106085, 12.19563478936745304193479045732, 12.65621841923551100758798437345, 13.55593111315306706842237051223

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