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Properties

Label 4-51e2-1.1-c1e2-0-2
Degree $4$
Conductor $2601$
Sign $1$
Analytic cond. $0.165842$
Root an. cond. $0.638151$
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  − 3·4-s + 8·7-s − 3·9-s − 4·13-s + 5·16-s − 8·19-s − 6·25-s − 24·28-s + 8·31-s + 9·36-s − 4·37-s + 8·43-s + 34·49-s + 12·52-s − 20·61-s − 24·63-s − 3·64-s + 8·67-s − 12·73-s + 24·76-s + 24·79-s + 9·81-s − 32·91-s + 4·97-s + 18·100-s + 16·103-s + 12·109-s + ⋯
L(s)  = 1  − 3/2·4-s + 3.02·7-s − 9-s − 1.10·13-s + 5/4·16-s − 1.83·19-s − 6/5·25-s − 4.53·28-s + 1.43·31-s + 3/2·36-s − 0.657·37-s + 1.21·43-s + 34/7·49-s + 1.66·52-s − 2.56·61-s − 3.02·63-s − 3/8·64-s + 0.977·67-s − 1.40·73-s + 2.75·76-s + 2.70·79-s + 81-s − 3.35·91-s + 0.406·97-s + 9/5·100-s + 1.57·103-s + 1.14·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2601\)    =    \(3^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(0.165842\)
Root analytic conductor: \(0.638151\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2601,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6131282898\)
\(L(\frac12)\) \(\approx\) \(0.6131282898\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2$ \( 1 + p T^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−13.13887678046194086016444609016, −12.16602218247269112718036034946, −11.93623114442209313196485845061, −11.11422065032987731085727997885, −10.71734099889110000433050476633, −9.912917364966795607879749961117, −8.872689562844898968412116321231, −8.695686711870280818326611584019, −7.81910395523808377988054564239, −7.81886564473676875900189501384, −6.12380991478521668699719766019, −5.15238580447119340700000765704, −4.74199315541377008016560012773, −4.18976898855068708054002343552, −2.14503914986953664301144200995, 2.14503914986953664301144200995, 4.18976898855068708054002343552, 4.74199315541377008016560012773, 5.15238580447119340700000765704, 6.12380991478521668699719766019, 7.81886564473676875900189501384, 7.81910395523808377988054564239, 8.695686711870280818326611584019, 8.872689562844898968412116321231, 9.912917364966795607879749961117, 10.71734099889110000433050476633, 11.11422065032987731085727997885, 11.93623114442209313196485845061, 12.16602218247269112718036034946, 13.13887678046194086016444609016

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