Properties

Label 4-185e2-1.1-c1e2-0-0
Degree $4$
Conductor $34225$
Sign $1$
Analytic cond. $2.18221$
Root an. cond. $1.21541$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·3-s − 4-s − 2·5-s − 4·6-s − 6·7-s + 8·8-s + 2·9-s + 4·10-s − 2·12-s + 4·13-s + 12·14-s − 4·15-s − 7·16-s − 4·18-s + 6·19-s + 2·20-s − 12·21-s − 16·23-s + 16·24-s − 25-s − 8·26-s + 6·27-s + 6·28-s − 14·29-s + 8·30-s + 6·31-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s − 1/2·4-s − 0.894·5-s − 1.63·6-s − 2.26·7-s + 2.82·8-s + 2/3·9-s + 1.26·10-s − 0.577·12-s + 1.10·13-s + 3.20·14-s − 1.03·15-s − 7/4·16-s − 0.942·18-s + 1.37·19-s + 0.447·20-s − 2.61·21-s − 3.33·23-s + 3.26·24-s − 1/5·25-s − 1.56·26-s + 1.15·27-s + 1.13·28-s − 2.59·29-s + 1.46·30-s + 1.07·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 34225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(34225\)    =    \(5^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(2.18221\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 34225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3530415589\)
\(L(\frac12)\) \(\approx\) \(0.3530415589\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 2 T + p T^{2} \)
37$C_2$ \( 1 + 2 T + p T^{2} \)
good2$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.2.c_f
3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_c
7$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.7.g_s
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.11.a_as
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
19$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.19.ag_s
23$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.23.q_eg
29$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.o_du
31$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.31.ag_s
41$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.41.a_ade
43$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.43.ay_iw
47$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.47.ak_by
53$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_s
59$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.59.ao_du
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.ac_c
67$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.67.ag_s
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.73.ac_c
79$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.79.ag_s
83$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.83.k_by
89$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.89.ak_by
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.a_afa
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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

−12.83700396025873518719060313011, −12.53421607436119648825260226143, −11.99992072438835276443555855726, −11.08622767691944824133600431390, −10.62904111038075452920472992408, −9.886485680215773428715856678466, −9.680822224664481339355704274917, −9.481537583834271805720213323745, −8.862834497522444094829440373628, −8.465687753732322302234960146005, −7.80613740742579314166288228579, −7.68277854882150940198308918500, −7.04349833318338023985203545872, −6.06537701611764299515641395379, −5.59419844091067508860520241629, −4.18449371322501977599649225864, −3.72657409626789884374060522575, −3.67341265117217924705438134962, −2.29561026262457437442647611969, −0.65300052429205403887487345028, 0.65300052429205403887487345028, 2.29561026262457437442647611969, 3.67341265117217924705438134962, 3.72657409626789884374060522575, 4.18449371322501977599649225864, 5.59419844091067508860520241629, 6.06537701611764299515641395379, 7.04349833318338023985203545872, 7.68277854882150940198308918500, 7.80613740742579314166288228579, 8.465687753732322302234960146005, 8.862834497522444094829440373628, 9.481537583834271805720213323745, 9.680822224664481339355704274917, 9.886485680215773428715856678466, 10.62904111038075452920472992408, 11.08622767691944824133600431390, 11.99992072438835276443555855726, 12.53421607436119648825260226143, 12.83700396025873518719060313011

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