Properties

Label 4-2850e2-1.1-c1e2-0-23
Degree $4$
Conductor $8122500$
Sign $1$
Analytic cond. $517.897$
Root an. cond. $4.77046$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 9-s − 8·11-s + 16-s + 2·19-s − 12·29-s + 8·31-s + 36-s − 4·41-s + 8·44-s − 2·49-s − 20·61-s − 64-s − 16·71-s − 2·76-s + 24·79-s + 81-s − 12·89-s + 8·99-s − 20·101-s − 20·109-s + 12·116-s + 26·121-s − 8·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s − 2.41·11-s + 1/4·16-s + 0.458·19-s − 2.22·29-s + 1.43·31-s + 1/6·36-s − 0.624·41-s + 1.20·44-s − 2/7·49-s − 2.56·61-s − 1/8·64-s − 1.89·71-s − 0.229·76-s + 2.70·79-s + 1/9·81-s − 1.27·89-s + 0.804·99-s − 1.99·101-s − 1.91·109-s + 1.11·116-s + 2.36·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8122500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(517.897\)
Root analytic conductor: \(4.77046\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2850} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 8122500,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
19$C_1$ \( ( 1 - T )^{2} \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.451511128489238999481231837066, −8.126303053854613912727024503382, −7.87555796351561730746783261132, −7.66507756362397417007507037886, −7.18179038001338540005834553364, −6.77109525572747945684145677023, −6.16187405108874303998107610623, −5.82445297933609728689010748376, −5.43767656219361214557493061118, −5.16364679086258722707088715683, −4.75142835684312780904009923070, −4.38442308749107546017678643428, −3.72406223689149617678799610208, −3.34524898848047897050540500432, −2.66952053907176113346416064893, −2.65896072812898951889515664551, −1.83021626033069968787825827153, −1.20111902159895274097960434072, 0, 0, 1.20111902159895274097960434072, 1.83021626033069968787825827153, 2.65896072812898951889515664551, 2.66952053907176113346416064893, 3.34524898848047897050540500432, 3.72406223689149617678799610208, 4.38442308749107546017678643428, 4.75142835684312780904009923070, 5.16364679086258722707088715683, 5.43767656219361214557493061118, 5.82445297933609728689010748376, 6.16187405108874303998107610623, 6.77109525572747945684145677023, 7.18179038001338540005834553364, 7.66507756362397417007507037886, 7.87555796351561730746783261132, 8.126303053854613912727024503382, 8.451511128489238999481231837066

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