Properties

Label 4-3450e2-1.1-c1e2-0-10
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 − 2·11-s + 16-s + 10·19-s + 18·29-s − 4·31-s + 36-s − 10·41-s + 2·44-s + 5·49-s + 24·59-s + 8·61-s − 64-s − 20·71-s − 10·76-s − 18·79-s + 81-s − 20·89-s + 2·99-s − 12·101-s + 32·109-s − 18·116-s − 19·121-s + 4·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s − 0.603·11-s + 1/4·16-s + 2.29·19-s + 3.34·29-s − 0.718·31-s + 1/6·36-s − 1.56·41-s + 0.301·44-s + 5/7·49-s + 3.12·59-s + 1.02·61-s − 1/8·64-s − 2.37·71-s − 1.14·76-s − 2.02·79-s + 1/9·81-s − 2.11·89-s + 0.201·99-s − 1.19·101-s + 3.06·109-s − 1.67·116-s − 1.72·121-s + 0.359·124-s + 0.0887·127-s + 0.0873·131-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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 11902500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.072275567\)
\(L(\frac12)\) \(\approx\) \(2.072275567\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 121 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
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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

−8.693097176556994973723218204531, −8.362142168317167204323087199359, −8.356544120925116181311029121788, −7.45113523505256771878113135146, −7.40123202268828390077788543279, −7.09919200682362263971655647303, −6.45286629265679600471550040984, −6.25256031595554610636956925277, −5.52630335117862415004986785902, −5.38090129497817072120354534357, −5.05510825016096281046454997916, −4.71430420188034048842501914865, −3.93837669139835754410187282357, −3.92017881302827089084914482572, −2.98971855228610307269385516107, −2.94225665659572919904413049886, −2.52659468178526845434472247657, −1.58727947131694748644038429790, −1.10779256574591983254957332394, −0.50394533001127800677203345637, 0.50394533001127800677203345637, 1.10779256574591983254957332394, 1.58727947131694748644038429790, 2.52659468178526845434472247657, 2.94225665659572919904413049886, 2.98971855228610307269385516107, 3.92017881302827089084914482572, 3.93837669139835754410187282357, 4.71430420188034048842501914865, 5.05510825016096281046454997916, 5.38090129497817072120354534357, 5.52630335117862415004986785902, 6.25256031595554610636956925277, 6.45286629265679600471550040984, 7.09919200682362263971655647303, 7.40123202268828390077788543279, 7.45113523505256771878113135146, 8.356544120925116181311029121788, 8.362142168317167204323087199359, 8.693097176556994973723218204531

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