Properties

Label 4-2850e2-1.1-c1e2-0-3
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 $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 9-s + 2·11-s + 16-s + 2·19-s − 6·29-s − 10·31-s + 36-s − 4·41-s − 2·44-s + 14·49-s − 22·61-s − 64-s + 4·71-s − 2·76-s − 34·79-s + 81-s − 14·89-s − 2·99-s − 16·101-s − 12·109-s + 6·116-s − 19·121-s + 10·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s + 0.603·11-s + 1/4·16-s + 0.458·19-s − 1.11·29-s − 1.79·31-s + 1/6·36-s − 0.624·41-s − 0.301·44-s + 2·49-s − 2.81·61-s − 1/8·64-s + 0.474·71-s − 0.229·76-s − 3.82·79-s + 1/9·81-s − 1.48·89-s − 0.201·99-s − 1.59·101-s − 1.14·109-s + 0.557·116-s − 1.72·121-s + 0.898·124-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: \(0\)
Selberg data: \((4,\ 8122500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7947045465\)
\(L(\frac12)\) \(\approx\) \(0.7947045465\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 133 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 137 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 17 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 157 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−9.319685545573095156782251187860, −8.557966385680972589698618709678, −8.435769313011631165278233580013, −7.70251644911617365019374728121, −7.54011624280925526415780954553, −7.05063172385998476454132587082, −6.90706050980109179698988188690, −6.12708863969531851006319969914, −5.91384900513542665602240656917, −5.38732582572456385147962861786, −5.34694383631505401796429914047, −4.59315042880377684866519876149, −4.18519511194129856097562017301, −3.88267159542914329155661524632, −3.40905153845812868487171191736, −2.90577199714807317140501161795, −2.44201825057909405270727932409, −1.49984459642685948920931233809, −1.47926245411887905522198437659, −0.28934424354243803265461894676, 0.28934424354243803265461894676, 1.47926245411887905522198437659, 1.49984459642685948920931233809, 2.44201825057909405270727932409, 2.90577199714807317140501161795, 3.40905153845812868487171191736, 3.88267159542914329155661524632, 4.18519511194129856097562017301, 4.59315042880377684866519876149, 5.34694383631505401796429914047, 5.38732582572456385147962861786, 5.91384900513542665602240656917, 6.12708863969531851006319969914, 6.90706050980109179698988188690, 7.05063172385998476454132587082, 7.54011624280925526415780954553, 7.70251644911617365019374728121, 8.435769313011631165278233580013, 8.557966385680972589698618709678, 9.319685545573095156782251187860

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