Properties

Label 4-3450e2-1.1-c1e2-0-7
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 − 8·11-s + 16-s − 8·19-s + 12·29-s − 16·31-s + 36-s + 20·41-s + 8·44-s + 14·49-s + 20·61-s − 64-s + 16·71-s + 8·76-s + 24·79-s + 81-s + 4·89-s + 8·99-s + 36·101-s − 4·109-s − 12·116-s + 26·121-s + 16·124-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 − 1.83·19-s + 2.22·29-s − 2.87·31-s + 1/6·36-s + 3.12·41-s + 1.20·44-s + 2·49-s + 2.56·61-s − 1/8·64-s + 1.89·71-s + 0.917·76-s + 2.70·79-s + 1/9·81-s + 0.423·89-s + 0.804·99-s + 3.58·101-s − 0.383·109-s − 1.11·116-s + 2.36·121-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

\(L(1)\) \(\approx\) \(1.414391133\)
\(L(\frac12)\) \(\approx\) \(1.414391133\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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

−8.747254709588423577340688029138, −8.342132617466661328431549257670, −8.081879826724826686626155483006, −7.77559413377405685528723924786, −7.30727041305469931317155355638, −7.11860118917362336569624195483, −6.39231239333070054480452787168, −6.17295461297151663563365913393, −5.66865917223226640709857301433, −5.35354256348786075248920017825, −5.01298742914042256728943913909, −4.67681722182461558600216132939, −4.08525513177826294622412090617, −3.80194338517480922719044684435, −3.28480431148627671517120141173, −2.55333156865305143807481482262, −2.24640206005644098778164929489, −2.19928422557990439415273858048, −0.823183893825295926897108568746, −0.47854328850114826098390521773, 0.47854328850114826098390521773, 0.823183893825295926897108568746, 2.19928422557990439415273858048, 2.24640206005644098778164929489, 2.55333156865305143807481482262, 3.28480431148627671517120141173, 3.80194338517480922719044684435, 4.08525513177826294622412090617, 4.67681722182461558600216132939, 5.01298742914042256728943913909, 5.35354256348786075248920017825, 5.66865917223226640709857301433, 6.17295461297151663563365913393, 6.39231239333070054480452787168, 7.11860118917362336569624195483, 7.30727041305469931317155355638, 7.77559413377405685528723924786, 8.081879826724826686626155483006, 8.342132617466661328431549257670, 8.747254709588423577340688029138

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