Properties

Label 4-455e2-1.1-c1e2-0-0
Degree $4$
Conductor $207025$
Sign $1$
Analytic cond. $13.2000$
Root an. cond. $1.90609$
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  + 2·3-s − 4·4-s − 3·9-s − 8·12-s + 5·13-s + 12·16-s + 6·17-s − 12·23-s + 25-s − 14·27-s + 6·29-s + 12·36-s + 10·39-s − 20·43-s + 24·48-s + 49-s + 12·51-s − 20·52-s + 24·53-s + 16·61-s − 32·64-s − 24·68-s − 24·69-s + 2·75-s − 2·79-s − 4·81-s + 12·87-s + ⋯
L(s)  = 1  + 1.15·3-s − 2·4-s − 9-s − 2.30·12-s + 1.38·13-s + 3·16-s + 1.45·17-s − 2.50·23-s + 1/5·25-s − 2.69·27-s + 1.11·29-s + 2·36-s + 1.60·39-s − 3.04·43-s + 3.46·48-s + 1/7·49-s + 1.68·51-s − 2.77·52-s + 3.29·53-s + 2.04·61-s − 4·64-s − 2.91·68-s − 2.88·69-s + 0.230·75-s − 0.225·79-s − 4/9·81-s + 1.28·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(207025\)    =    \(5^{2} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(13.2000\)
Root analytic conductor: \(1.90609\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{207025} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 207025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.233308737\)
\(L(\frac12)\) \(\approx\) \(1.233308737\)
\(L(\frac{3}{2})\) 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$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_2$ \( 1 - 5 T + p T^{2} \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
3$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
97$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
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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

−8.679498853252053444462430241617, −8.655493509958517556104692087607, −8.378748037315465445192790829210, −7.980947296320422939728903060714, −7.58370080180617520189810021223, −6.52421319630927198344725128175, −5.95302503292927073396566116958, −5.47907564040983295563221596641, −5.25579521602379928421249093772, −4.29829876881027420494319425466, −3.66964895044272310843171490096, −3.61689003045514298236484607491, −2.92042524544419550067578696465, −1.88740825483186720696747488093, −0.68446346217697022883479913777, 0.68446346217697022883479913777, 1.88740825483186720696747488093, 2.92042524544419550067578696465, 3.61689003045514298236484607491, 3.66964895044272310843171490096, 4.29829876881027420494319425466, 5.25579521602379928421249093772, 5.47907564040983295563221596641, 5.95302503292927073396566116958, 6.52421319630927198344725128175, 7.58370080180617520189810021223, 7.980947296320422939728903060714, 8.378748037315465445192790829210, 8.655493509958517556104692087607, 8.679498853252053444462430241617

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