Properties

Label 4-3450e2-1.1-c1e2-0-21
Degree $4$
Conductor $11902500$
Sign $1$
Analytic cond. $758.913$
Root an. cond. $5.24865$
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 + 4·11-s + 16-s − 16·19-s + 20·29-s + 16·31-s + 36-s − 12·41-s − 4·44-s + 10·49-s − 8·59-s + 24·61-s − 64-s + 32·71-s + 16·76-s − 20·79-s + 81-s − 4·99-s + 20·101-s + 16·109-s − 20·116-s − 10·121-s − 16·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s + 1.20·11-s + 1/4·16-s − 3.67·19-s + 3.71·29-s + 2.87·31-s + 1/6·36-s − 1.87·41-s − 0.603·44-s + 10/7·49-s − 1.04·59-s + 3.07·61-s − 1/8·64-s + 3.79·71-s + 1.83·76-s − 2.25·79-s + 1/9·81-s − 0.402·99-s + 1.99·101-s + 1.53·109-s − 1.85·116-s − 0.909·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(11902500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(758.913\)
Root analytic conductor: \(5.24865\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3450} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 11902500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.722868171\)
\(L(\frac12)\) \(\approx\) \(2.722868171\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23$C_2$ \( 1 + T^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
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

−8.491725685986546108368650146300, −8.453679605485522263187435528469, −8.281028368073281694480300016272, −8.049624222480126240166021194579, −7.01693371565295696275902382588, −6.74817882688283841819901377021, −6.53160607052570787030599976959, −6.35620876400494674680637922228, −5.97046187158078084221185552061, −5.21848510036479502162006803669, −4.92166332292426082433719553744, −4.37144490027030198185244195611, −4.21041166663859231026408996133, −4.02687175180789504037043217108, −3.00778413161477168460644940239, −2.99801721752752521504811295188, −2.11006488667088853878591070086, −1.99294127676939544676961041320, −0.77961067081901145361420462777, −0.74854409572395087390692142387, 0.74854409572395087390692142387, 0.77961067081901145361420462777, 1.99294127676939544676961041320, 2.11006488667088853878591070086, 2.99801721752752521504811295188, 3.00778413161477168460644940239, 4.02687175180789504037043217108, 4.21041166663859231026408996133, 4.37144490027030198185244195611, 4.92166332292426082433719553744, 5.21848510036479502162006803669, 5.97046187158078084221185552061, 6.35620876400494674680637922228, 6.53160607052570787030599976959, 6.74817882688283841819901377021, 7.01693371565295696275902382588, 8.049624222480126240166021194579, 8.281028368073281694480300016272, 8.453679605485522263187435528469, 8.491725685986546108368650146300

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