Properties

Label 4-1950e2-1.1-c1e2-0-42
Degree $4$
Conductor $3802500$
Sign $1$
Analytic cond. $242.450$
Root an. cond. $3.94598$
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-s + 3-s − 6-s + 2·7-s + 8-s + 3·11-s − 2·13-s − 2·14-s − 16-s + 6·17-s − 2·19-s + 2·21-s − 3·22-s + 3·23-s + 24-s + 2·26-s − 27-s − 3·29-s + 10·31-s + 3·33-s − 6·34-s − 7·37-s + 2·38-s − 2·39-s − 6·41-s − 2·42-s − 43-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 0.904·11-s − 0.554·13-s − 0.534·14-s − 1/4·16-s + 1.45·17-s − 0.458·19-s + 0.436·21-s − 0.639·22-s + 0.625·23-s + 0.204·24-s + 0.392·26-s − 0.192·27-s − 0.557·29-s + 1.79·31-s + 0.522·33-s − 1.02·34-s − 1.15·37-s + 0.324·38-s − 0.320·39-s − 0.937·41-s − 0.308·42-s − 0.152·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3802500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(242.450\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1950} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3802500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.584466924\)
\(L(\frac12)\) \(\approx\) \(2.584466924\)
\(L(\frac{3}{2})\) 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)$
bad2$C_2$ \( 1 + T + T^{2} \)
3$C_2$ \( 1 - T + T^{2} \)
5 \( 1 \)
13$C_2$ \( 1 + 2 T + p T^{2} \)
good7$C_2^2$ \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 12 T + 73 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
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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

−9.288148940744875302903842409668, −8.855687079027789533635019197924, −8.634161964045267640506025263673, −8.357922371685509524913261117338, −7.88107534437379347561434113629, −7.57591604390623233412175286357, −7.07875357228621152288227315050, −6.87959660604213375400015393669, −6.33195955562865746099882395958, −5.61415099124176273504745045174, −5.52698662445578510297768216543, −4.81673672489245849860941981419, −4.54483583737846357748809669659, −3.97126716701134649686013872680, −3.43967941462919645926431329884, −3.14089040403279847271914274533, −2.19705926601569468319071382308, −2.07712842150609202458567023617, −1.14886637835919495862437369097, −0.74995686453470973490792882461, 0.74995686453470973490792882461, 1.14886637835919495862437369097, 2.07712842150609202458567023617, 2.19705926601569468319071382308, 3.14089040403279847271914274533, 3.43967941462919645926431329884, 3.97126716701134649686013872680, 4.54483583737846357748809669659, 4.81673672489245849860941981419, 5.52698662445578510297768216543, 5.61415099124176273504745045174, 6.33195955562865746099882395958, 6.87959660604213375400015393669, 7.07875357228621152288227315050, 7.57591604390623233412175286357, 7.88107534437379347561434113629, 8.357922371685509524913261117338, 8.634161964045267640506025263673, 8.855687079027789533635019197924, 9.288148940744875302903842409668

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