Properties

Label 4-642816-1.1-c1e2-0-0
Degree $4$
Conductor $642816$
Sign $1$
Analytic cond. $40.9865$
Root an. cond. $2.53023$
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  − 4·19-s + 6·25-s − 3·31-s + 4·37-s − 4·43-s − 14·49-s + 4·61-s + 4·67-s + 8·73-s + 12·79-s + 4·97-s − 12·103-s + 4·109-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  − 0.917·19-s + 6/5·25-s − 0.538·31-s + 0.657·37-s − 0.609·43-s − 2·49-s + 0.512·61-s + 0.488·67-s + 0.936·73-s + 1.35·79-s + 0.406·97-s − 1.18·103-s + 0.383·109-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

\[\begin{aligned}\Lambda(s)=\mathstrut & 642816 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 642816 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(642816\)    =    \(2^{8} \cdot 3^{4} \cdot 31\)
Sign: $1$
Analytic conductor: \(40.9865\)
Root analytic conductor: \(2.53023\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{642816} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 642816,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.672635649\)
\(L(\frac12)\) \(\approx\) \(1.672635649\)
\(L(\frac{3}{2})\) 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 \( 1 \)
31$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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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

−8.287491373991491340361830694568, −8.020318665753716063498731364642, −7.54944714850268034982134549083, −6.85906047742591830452567521410, −6.58569303925538710605927521470, −6.29937347033500949336656356897, −5.40589591964885115337945892747, −5.30652527345799773906641983110, −4.47801590330248449167427672400, −4.29061007211634116317389406613, −3.40308279207134752465432209555, −3.09807841848495628534010519689, −2.25518327350493679823246657235, −1.70627062959602532269086841198, −0.65190739677216990524977625422, 0.65190739677216990524977625422, 1.70627062959602532269086841198, 2.25518327350493679823246657235, 3.09807841848495628534010519689, 3.40308279207134752465432209555, 4.29061007211634116317389406613, 4.47801590330248449167427672400, 5.30652527345799773906641983110, 5.40589591964885115337945892747, 6.29937347033500949336656356897, 6.58569303925538710605927521470, 6.85906047742591830452567521410, 7.54944714850268034982134549083, 8.020318665753716063498731364642, 8.287491373991491340361830694568

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