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Properties

Label 4-650e2-1.1-c1e2-0-26
Degree $4$
Conductor $422500$
Sign $-1$
Analytic cond. $26.9389$
Root an. cond. $2.27821$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s + 4·7-s − 4·8-s − 5·9-s − 4·13-s − 8·14-s + 5·16-s + 10·18-s + 8·26-s + 12·28-s − 6·32-s − 15·36-s + 4·37-s + 24·47-s − 2·49-s − 12·52-s − 16·56-s + 4·61-s − 20·63-s + 7·64-s − 26·67-s + 20·72-s + 22·73-s − 8·74-s − 20·79-s + 16·81-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 1.51·7-s − 1.41·8-s − 5/3·9-s − 1.10·13-s − 2.13·14-s + 5/4·16-s + 2.35·18-s + 1.56·26-s + 2.26·28-s − 1.06·32-s − 5/2·36-s + 0.657·37-s + 3.50·47-s − 2/7·49-s − 1.66·52-s − 2.13·56-s + 0.512·61-s − 2.51·63-s + 7/8·64-s − 3.17·67-s + 2.35·72-s + 2.57·73-s − 0.929·74-s − 2.25·79-s + 16/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 422500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 422500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(422500\)    =    \(2^{2} \cdot 5^{4} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(26.9389\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 422500,\ (\ :1/2, 1/2),\ -1)\)

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 $\mathbf{F}_p$
bad2$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
13$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.3.a_f
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.7.ae_s
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.11.a_n
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.a_z
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$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.31.a_cg
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.41.a_cv
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.47.ay_je
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.67.ba_lr
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.73.aw_kh
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.79.u_jy
83$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.83.s_jn
89$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.89.a_abv
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−8.316812877075225116243141130520, −8.205833330580628456828191862699, −7.53930750814145995616688594594, −7.29853134793764088505009214313, −6.79304570165130919460023613639, −5.88282891686090259551787479295, −5.79711527111474135586035057131, −5.20592148788576143134616479765, −4.59559413232058600408784127991, −3.99467974936225470352105085052, −2.90586413258629518583139678045, −2.65155040417235216240951299295, −1.98306192934560204923948985995, −1.14126627842990540054683071421, 0, 1.14126627842990540054683071421, 1.98306192934560204923948985995, 2.65155040417235216240951299295, 2.90586413258629518583139678045, 3.99467974936225470352105085052, 4.59559413232058600408784127991, 5.20592148788576143134616479765, 5.79711527111474135586035057131, 5.88282891686090259551787479295, 6.79304570165130919460023613639, 7.29853134793764088505009214313, 7.53930750814145995616688594594, 8.205833330580628456828191862699, 8.316812877075225116243141130520

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