Properties

Label 4-325e2-1.1-c1e2-0-19
Degree $4$
Conductor $105625$
Sign $1$
Analytic cond. $6.73474$
Root an. cond. $1.61094$
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  + 4·3-s + 3·4-s + 6·9-s + 12·12-s − 4·13-s + 5·16-s + 12·23-s − 4·27-s − 12·29-s + 18·36-s − 16·39-s − 12·43-s + 20·48-s + 14·49-s − 12·52-s − 24·53-s + 12·61-s + 3·64-s + 48·69-s − 37·81-s − 48·87-s + 36·92-s + 12·101-s + 12·103-s + 12·107-s − 12·108-s − 36·116-s + ⋯
L(s)  = 1  + 2.30·3-s + 3/2·4-s + 2·9-s + 3.46·12-s − 1.10·13-s + 5/4·16-s + 2.50·23-s − 0.769·27-s − 2.22·29-s + 3·36-s − 2.56·39-s − 1.82·43-s + 2.88·48-s + 2·49-s − 1.66·52-s − 3.29·53-s + 1.53·61-s + 3/8·64-s + 5.77·69-s − 4.11·81-s − 5.14·87-s + 3.75·92-s + 1.19·101-s + 1.18·103-s + 1.16·107-s − 1.15·108-s − 3.34·116-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(105625\)    =    \(5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(6.73474\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{325} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 105625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.288361690\)
\(L(\frac12)\) \(\approx\) \(4.288361690\)
\(L(\frac{3}{2})\) 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)$
bad5 \( 1 \)
13$C_2$ \( 1 + 4 T + p T^{2} \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 138 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 158 T^{2} + p^{2} T^{4} \)
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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

−11.51097974961019453693761398593, −11.48335480489018205382092358843, −11.04126496168835771267326474181, −10.38546052576691879232312697168, −9.793977656800040321730176468921, −9.432934503057887530189474089897, −8.886620010724159412422674149030, −8.723785406429474636325079626613, −7.83657285223277931890250951501, −7.72138239672211531780943250613, −7.14920297447368846528732085139, −6.91087513343003390718548320993, −6.09016604512008503144563533649, −5.39606012788722682752404915554, −4.79770767370664352185513698856, −3.73961609501620932225141553181, −3.20114426675046430668429134682, −2.90609290108933831304524676338, −2.19968394585130162988212608852, −1.77144920368671874843295631541, 1.77144920368671874843295631541, 2.19968394585130162988212608852, 2.90609290108933831304524676338, 3.20114426675046430668429134682, 3.73961609501620932225141553181, 4.79770767370664352185513698856, 5.39606012788722682752404915554, 6.09016604512008503144563533649, 6.91087513343003390718548320993, 7.14920297447368846528732085139, 7.72138239672211531780943250613, 7.83657285223277931890250951501, 8.723785406429474636325079626613, 8.886620010724159412422674149030, 9.432934503057887530189474089897, 9.793977656800040321730176468921, 10.38546052576691879232312697168, 11.04126496168835771267326474181, 11.48335480489018205382092358843, 11.51097974961019453693761398593

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