Properties

Label 4-1881e2-1.1-c1e2-0-2
Degree $4$
Conductor $3538161$
Sign $-1$
Analytic cond. $225.596$
Root an. cond. $3.87554$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·4-s + 5·16-s − 10·25-s − 8·31-s + 4·37-s − 2·49-s − 3·64-s − 16·67-s + 36·97-s + 30·100-s + 8·103-s − 11·121-s + 24·124-s + 127-s + 131-s + 137-s + 139-s − 12·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 3/2·4-s + 5/4·16-s − 2·25-s − 1.43·31-s + 0.657·37-s − 2/7·49-s − 3/8·64-s − 1.95·67-s + 3.65·97-s + 3·100-s + 0.788·103-s − 121-s + 2.15·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.986·148-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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3538161 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3538161 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3538161\)    =    \(3^{4} \cdot 11^{2} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(225.596\)
Root analytic conductor: \(3.87554\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 3538161,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
11$C_2$ \( 1 + p T^{2} \)
19$C_2$ \( 1 + T^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
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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

−7.45300607607148310732730641518, −6.93996937508449309694153622374, −6.21890883232615600393019819513, −6.04844283405795656186748401507, −5.50002341673501801932282313196, −5.21555075922375481109088525130, −4.63793871370218187298161565920, −4.30047605629390431556630160676, −3.89910118576627159283110673794, −3.45998670923653187818882160342, −2.97857023525784756750413936323, −2.09443376757349078468909765521, −1.70261064264205472030898492729, −0.72197054467804852484866678869, 0, 0.72197054467804852484866678869, 1.70261064264205472030898492729, 2.09443376757349078468909765521, 2.97857023525784756750413936323, 3.45998670923653187818882160342, 3.89910118576627159283110673794, 4.30047605629390431556630160676, 4.63793871370218187298161565920, 5.21555075922375481109088525130, 5.50002341673501801932282313196, 6.04844283405795656186748401507, 6.21890883232615600393019819513, 6.93996937508449309694153622374, 7.45300607607148310732730641518

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