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Properties

Label 4-650e2-1.1-c1e2-0-32
Degree $4$
Conductor $422500$
Sign $-1$
Analytic cond. $26.9389$
Root an. cond. $2.27821$
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  + 2·2-s + 2·4-s − 2·7-s − 9-s + 2·13-s − 4·14-s − 4·16-s − 2·18-s + 4·26-s − 4·28-s + 2·29-s − 8·32-s − 2·36-s − 14·37-s + 4·47-s − 10·49-s + 4·52-s + 4·58-s − 18·61-s + 2·63-s − 8·64-s + 4·67-s + 2·73-s − 28·74-s − 8·81-s + 22·83-s − 4·91-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 0.755·7-s − 1/3·9-s + 0.554·13-s − 1.06·14-s − 16-s − 0.471·18-s + 0.784·26-s − 0.755·28-s + 0.371·29-s − 1.41·32-s − 1/3·36-s − 2.30·37-s + 0.583·47-s − 1.42·49-s + 0.554·52-s + 0.525·58-s − 2.30·61-s + 0.251·63-s − 64-s + 0.488·67-s + 0.234·73-s − 3.25·74-s − 8/9·81-s + 2.41·83-s − 0.419·91-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: yes
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$C_2$ \( 1 - p T + p T^{2} \)
5 \( 1 \)
13$C_2$ \( 1 - 2 T + p T^{2} \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.3.a_b
7$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.c_o
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.11.a_ag
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.19.a_c
23$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \) 2.23.a_al
29$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.29.ac_af
31$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.31.a_cg
37$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.o_ek
41$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.41.a_k
43$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.43.a_bl
47$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.47.ae_de
53$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \) 2.53.a_acn
59$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.59.a_ba
61$C_2$$\times$$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.61.s_hr
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.67.ae_fe
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.71.a_c
73$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.ac_ada
79$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.79.a_ef
83$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.83.aw_kc
89$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.89.a_by
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.97.c_he
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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

−8.516558842801950142289185772881, −7.82276303139833657289307966146, −7.30546396785080265560669481863, −6.75001764062555649092575302744, −6.38177852467008027292123594822, −6.05791652548158987419982566771, −5.50189857012046315781588006655, −4.97949533292399938588775089019, −4.64334862231214422049307834811, −3.81042352534998735544297567969, −3.52983690007913581069345114537, −3.03431952651838337062663671838, −2.40017972159807830216450523838, −1.51804489120054525868878559026, 0, 1.51804489120054525868878559026, 2.40017972159807830216450523838, 3.03431952651838337062663671838, 3.52983690007913581069345114537, 3.81042352534998735544297567969, 4.64334862231214422049307834811, 4.97949533292399938588775089019, 5.50189857012046315781588006655, 6.05791652548158987419982566771, 6.38177852467008027292123594822, 6.75001764062555649092575302744, 7.30546396785080265560669481863, 7.82276303139833657289307966146, 8.516558842801950142289185772881

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