Properties

Label 4-1152e2-1.1-c1e2-0-24
Degree $4$
Conductor $1327104$
Sign $1$
Analytic cond. $84.6173$
Root an. cond. $3.03294$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 6·9-s − 7·11-s + 2·17-s + 8·19-s + 7·25-s + 9·27-s − 21·33-s − 11·41-s + 43-s − 5·49-s + 6·51-s + 24·57-s − 59-s + 7·67-s + 14·73-s + 21·75-s + 9·81-s − 6·83-s + 12·89-s − 3·97-s − 42·99-s + 32·107-s − 14·113-s + 15·121-s − 33·123-s + 127-s + ⋯
L(s)  = 1  + 1.73·3-s + 2·9-s − 2.11·11-s + 0.485·17-s + 1.83·19-s + 7/5·25-s + 1.73·27-s − 3.65·33-s − 1.71·41-s + 0.152·43-s − 5/7·49-s + 0.840·51-s + 3.17·57-s − 0.130·59-s + 0.855·67-s + 1.63·73-s + 2.42·75-s + 81-s − 0.658·83-s + 1.27·89-s − 0.304·97-s − 4.22·99-s + 3.09·107-s − 1.31·113-s + 1.36·121-s − 2.97·123-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1327104\)    =    \(2^{14} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(84.6173\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1327104,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.021490067\)
\(L(\frac12)\) \(\approx\) \(4.021490067\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
good5$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2^2$ \( 1 + 9 T^{2} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \)
23$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 53 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 3 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 3 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 63 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−7.899030520394456735215973571712, −7.73566421574064553201839388534, −7.29241998862685358862473450646, −6.91296454232352804391096124581, −6.38540293133911611565558640611, −5.55617299690309204548939386447, −5.20778286759431769565513605570, −4.92897503593597242673462858950, −4.33866880353343236915641708850, −3.45279582029743327474111645990, −3.27746691205870178857374565883, −2.88816574598995870812108728148, −2.32175335116065284967742395307, −1.71692284008992558140237984995, −0.799693191645555365371420429636, 0.799693191645555365371420429636, 1.71692284008992558140237984995, 2.32175335116065284967742395307, 2.88816574598995870812108728148, 3.27746691205870178857374565883, 3.45279582029743327474111645990, 4.33866880353343236915641708850, 4.92897503593597242673462858950, 5.20778286759431769565513605570, 5.55617299690309204548939386447, 6.38540293133911611565558640611, 6.91296454232352804391096124581, 7.29241998862685358862473450646, 7.73566421574064553201839388534, 7.899030520394456735215973571712

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