Properties

Label 4-6300e2-1.1-c1e2-0-0
Degree $4$
Conductor $39690000$
Sign $1$
Analytic cond. $2530.66$
Root an. cond. $7.09265$
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  − 4·11-s − 12·19-s − 4·29-s + 20·31-s − 20·41-s − 49-s − 16·59-s + 12·61-s + 12·71-s + 16·79-s − 20·89-s − 12·101-s − 36·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 1.20·11-s − 2.75·19-s − 0.742·29-s + 3.59·31-s − 3.12·41-s − 1/7·49-s − 2.08·59-s + 1.53·61-s + 1.42·71-s + 1.80·79-s − 2.11·89-s − 1.19·101-s − 3.44·109-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(39690000\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(2530.66\)
Root analytic conductor: \(7.09265\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6300} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 39690000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.07580578033\)
\(L(\frac12)\) \(\approx\) \(0.07580578033\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−8.346828743525269137659479265272, −7.939915264148380675405234112440, −7.82055382401402016956848164593, −6.87101946947701337829544411602, −6.78859504872157359077603761929, −6.41033906921793711440143414599, −6.40352020443916823903318927364, −5.63030206803935319966843218590, −5.37674179666073660698854757516, −4.87022076872457111084152336141, −4.76098828910067998060959775398, −4.09519789922521185435075285690, −4.04315797698719992973885550375, −3.36954068535566271756527601226, −2.84960237846029342875821363530, −2.56760083606041036107231029371, −2.16895641289312599970610353966, −1.65426306973481533607872085843, −1.05135736658835141369752897040, −0.07234495905298095023881329306, 0.07234495905298095023881329306, 1.05135736658835141369752897040, 1.65426306973481533607872085843, 2.16895641289312599970610353966, 2.56760083606041036107231029371, 2.84960237846029342875821363530, 3.36954068535566271756527601226, 4.04315797698719992973885550375, 4.09519789922521185435075285690, 4.76098828910067998060959775398, 4.87022076872457111084152336141, 5.37674179666073660698854757516, 5.63030206803935319966843218590, 6.40352020443916823903318927364, 6.41033906921793711440143414599, 6.78859504872157359077603761929, 6.87101946947701337829544411602, 7.82055382401402016956848164593, 7.939915264148380675405234112440, 8.346828743525269137659479265272

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