Properties

Label 4-2850e2-1.1-c1e2-0-10
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 − 8·11-s + 16-s − 2·19-s + 20·29-s − 4·31-s + 36-s + 16·41-s + 8·44-s + 10·49-s + 4·59-s + 4·61-s − 64-s + 2·76-s + 4·79-s + 81-s + 24·89-s + 8·99-s + 12·101-s − 24·109-s − 20·116-s + 26·121-s + 4·124-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 − 0.458·19-s + 3.71·29-s − 0.718·31-s + 1/6·36-s + 2.49·41-s + 1.20·44-s + 10/7·49-s + 0.520·59-s + 0.512·61-s − 1/8·64-s + 0.229·76-s + 0.450·79-s + 1/9·81-s + 2.54·89-s + 0.804·99-s + 1.19·101-s − 2.29·109-s − 1.85·116-s + 2.36·121-s + 0.359·124-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

\(L(1)\) \(\approx\) \(1.784929403\)
\(L(\frac12)\) \(\approx\) \(1.784929403\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
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$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−8.919522091086955405478468955420, −8.486619549428608830459328639220, −8.148408168919827678960818082060, −8.009652851944364541376584549456, −7.57259251747135335320271882253, −7.20000570504533894710831757201, −6.58585457551800981860114958620, −6.38378191956660741830353379255, −5.76276361100783729090954801250, −5.44507793456338685945567395811, −5.19433612099057902841209979553, −4.61302521802055308166902626509, −4.40798568885981061719820156992, −3.92844086439447782908691440136, −3.08900901967576802620184400120, −2.91447135895081098722340698801, −2.42389629734007886126083555974, −2.06068917090482858792635915136, −0.860885018401932602538821588881, −0.57454389415014413932229875682, 0.57454389415014413932229875682, 0.860885018401932602538821588881, 2.06068917090482858792635915136, 2.42389629734007886126083555974, 2.91447135895081098722340698801, 3.08900901967576802620184400120, 3.92844086439447782908691440136, 4.40798568885981061719820156992, 4.61302521802055308166902626509, 5.19433612099057902841209979553, 5.44507793456338685945567395811, 5.76276361100783729090954801250, 6.38378191956660741830353379255, 6.58585457551800981860114958620, 7.20000570504533894710831757201, 7.57259251747135335320271882253, 8.009652851944364541376584549456, 8.148408168919827678960818082060, 8.486619549428608830459328639220, 8.919522091086955405478468955420

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