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Properties

Label 4-275e2-1.1-c1e2-0-14
Degree $4$
Conductor $75625$
Sign $1$
Analytic cond. $4.82191$
Root an. cond. $1.48185$
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  − 2-s − 3-s + 6-s − 5·7-s − 8-s − 2·9-s − 2·11-s − 10·13-s + 5·14-s − 16-s − 3·17-s + 2·18-s − 2·19-s + 5·21-s + 2·22-s + 11·23-s + 24-s + 10·26-s + 2·27-s − 9·29-s + 6·31-s + 6·32-s + 2·33-s + 3·34-s − 12·37-s + 2·38-s + 10·39-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.408·6-s − 1.88·7-s − 0.353·8-s − 2/3·9-s − 0.603·11-s − 2.77·13-s + 1.33·14-s − 1/4·16-s − 0.727·17-s + 0.471·18-s − 0.458·19-s + 1.09·21-s + 0.426·22-s + 2.29·23-s + 0.204·24-s + 1.96·26-s + 0.384·27-s − 1.67·29-s + 1.07·31-s + 1.06·32-s + 0.348·33-s + 0.514·34-s − 1.97·37-s + 0.324·38-s + 1.60·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(75625\)    =    \(5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(4.82191\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 75625,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) 2.2.b_b
3$D_{4}$ \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) 2.3.b_d
7$D_{4}$ \( 1 + 5 T + 17 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.7.f_r
13$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.13.k_bz
17$D_{4}$ \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_h
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.19.c_bn
23$D_{4}$ \( 1 - 11 T + 73 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.23.al_cv
29$D_{4}$ \( 1 + 9 T + 49 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.29.j_bx
31$D_{4}$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.31.ag_t
37$D_{4}$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.37.m_dt
41$D_{4}$ \( 1 + 4 T + 73 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.41.e_cv
43$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.43.a_bi
47$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.47.ag_dz
53$D_{4}$ \( 1 + T + 103 T^{2} + p T^{3} + p^{2} T^{4} \) 2.53.b_dz
59$D_{4}$ \( 1 + 14 T + 115 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.59.o_el
61$D_{4}$ \( 1 + 5 T + 99 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.61.f_dv
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$D_{4}$ \( 1 - 2 T + 130 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.71.ac_fa
73$D_{4}$ \( 1 - 5 T + 123 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.73.af_et
79$D_{4}$ \( 1 + 11 T + 159 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.79.l_gd
83$D_{4}$ \( 1 + 11 T + 115 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.83.l_el
89$D_{4}$ \( 1 - 7 T + 187 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.89.ah_hf
97$D_{4}$ \( 1 + 27 T + 373 T^{2} + 27 p T^{3} + p^{2} T^{4} \) 2.97.bb_oj
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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

−11.67020501557625141701675509837, −11.01779809857128900000577135767, −10.75770470419501364913570977747, −10.00820008617195022368519186425, −9.714380402320605627927132262200, −9.443182157871460324338093872423, −8.819873036028638500292269708527, −8.605904675749065979491153211911, −7.61296540418524370081381517166, −7.05460335950302119734931810000, −6.92346215499356070623832759682, −6.24236146899732046281076286266, −5.64031011464947950521292302591, −4.96521826308231203896098221582, −4.66495966213269380614410498206, −3.39743855585965294744355267594, −2.85320199008313282448555336254, −2.34628841478842226900834011508, 0, 0, 2.34628841478842226900834011508, 2.85320199008313282448555336254, 3.39743855585965294744355267594, 4.66495966213269380614410498206, 4.96521826308231203896098221582, 5.64031011464947950521292302591, 6.24236146899732046281076286266, 6.92346215499356070623832759682, 7.05460335950302119734931810000, 7.61296540418524370081381517166, 8.605904675749065979491153211911, 8.819873036028638500292269708527, 9.443182157871460324338093872423, 9.714380402320605627927132262200, 10.00820008617195022368519186425, 10.75770470419501364913570977747, 11.01779809857128900000577135767, 11.67020501557625141701675509837

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