Properties

Label 4-15e4-1.1-c3e2-0-1
Degree $4$
Conductor $50625$
Sign $1$
Analytic cond. $176.237$
Root an. cond. $3.64354$
Motivic weight $3$
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  − 64·11-s − 64·16-s − 200·19-s − 100·29-s − 216·31-s − 44·41-s + 650·49-s + 1.00e3·59-s − 1.03e3·61-s − 824·71-s − 1.20e3·79-s − 300·89-s − 1.40e3·101-s + 1.10e3·109-s + 410·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.95e3·169-s + 173-s + 4.09e3·176-s + ⋯
L(s)  = 1  − 1.75·11-s − 16-s − 2.41·19-s − 0.640·29-s − 1.25·31-s − 0.167·41-s + 1.89·49-s + 2.20·59-s − 2.17·61-s − 1.37·71-s − 1.70·79-s − 0.357·89-s − 1.38·101-s + 0.966·109-s + 0.308·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.34·169-s + 0.000439·173-s + 1.75·176-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(50625\)    =    \(3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(176.237\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{225} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 50625,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5026066779\)
\(L(\frac12)\) \(\approx\) \(0.5026066779\)
\(L(\frac{5}{2})\) 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)$
bad3 \( 1 \)
5 \( 1 \)
good2$C_2$ \( ( 1 - p^{2} T + p^{3} T^{2} )( 1 + p^{2} T + p^{3} T^{2} ) \)
7$C_2^2$ \( 1 - 650 T^{2} + p^{6} T^{4} \)
11$C_2$ \( ( 1 + 32 T + p^{3} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 2950 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 - 9150 T^{2} + p^{6} T^{4} \)
19$C_2$ \( ( 1 + 100 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 18250 T^{2} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 50 T + p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 108 T + p^{3} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 30550 T^{2} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 22 T + p^{3} T^{2} )^{2} \)
43$C_2^2$ \( 1 + 36350 T^{2} + p^{6} T^{4} \)
47$C_2^2$ \( 1 + 56550 T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 297750 T^{2} + p^{6} T^{4} \)
59$C_2$ \( ( 1 - 500 T + p^{3} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 518 T + p^{3} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 585650 T^{2} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 412 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 7150 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 + 600 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 1064050 T^{2} + p^{6} T^{4} \)
89$C_2$ \( ( 1 + 150 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 1676350 T^{2} + p^{6} T^{4} \)
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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

−12.66464164923169188111478288414, −11.48999159216930931950021529863, −10.91282225535868211705322114922, −10.53499610710502673366448436790, −10.42965859123453469199420682920, −9.555324731457588957912947919611, −9.085699558658319960723887401631, −8.383308826170656975080157514172, −8.362490277408615677388252496252, −7.32488124350269556007860738421, −7.21177933613361833415980915041, −6.43856895581418512275817067461, −5.69287594106284963827397012071, −5.43477343017973845487129109246, −4.37910657868203024511301877637, −4.29135622900123821162032689401, −3.15903330841775996080496884823, −2.37348008922852876239774577557, −1.91314019147042051215469236949, −0.27358447374549899257055219164, 0.27358447374549899257055219164, 1.91314019147042051215469236949, 2.37348008922852876239774577557, 3.15903330841775996080496884823, 4.29135622900123821162032689401, 4.37910657868203024511301877637, 5.43477343017973845487129109246, 5.69287594106284963827397012071, 6.43856895581418512275817067461, 7.21177933613361833415980915041, 7.32488124350269556007860738421, 8.362490277408615677388252496252, 8.383308826170656975080157514172, 9.085699558658319960723887401631, 9.555324731457588957912947919611, 10.42965859123453469199420682920, 10.53499610710502673366448436790, 10.91282225535868211705322114922, 11.48999159216930931950021529863, 12.66464164923169188111478288414

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