Properties

Label 4-100352-1.1-c1e2-0-17
Degree $4$
Conductor $100352$
Sign $1$
Analytic cond. $6.39853$
Root an. cond. $1.59045$
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  + 4·3-s + 6·9-s + 8·11-s − 4·17-s + 12·19-s − 10·25-s − 4·27-s + 32·33-s − 20·41-s − 8·43-s + 49-s − 16·51-s + 48·57-s − 20·59-s + 16·67-s − 12·73-s − 40·75-s − 37·81-s − 4·83-s + 36·89-s − 4·97-s + 48·99-s + 32·107-s + 12·113-s + 26·121-s − 80·123-s + 127-s + ⋯
L(s)  = 1  + 2.30·3-s + 2·9-s + 2.41·11-s − 0.970·17-s + 2.75·19-s − 2·25-s − 0.769·27-s + 5.57·33-s − 3.12·41-s − 1.21·43-s + 1/7·49-s − 2.24·51-s + 6.35·57-s − 2.60·59-s + 1.95·67-s − 1.40·73-s − 4.61·75-s − 4.11·81-s − 0.439·83-s + 3.81·89-s − 0.406·97-s + 4.82·99-s + 3.09·107-s + 1.12·113-s + 2.36·121-s − 7.21·123-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100352 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100352 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(100352\)    =    \(2^{11} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(6.39853\)
Root analytic conductor: \(1.59045\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 100352,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.528858290\)
\(L(\frac12)\) \(\approx\) \(3.528858290\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−9.389652658008486485429618465354, −9.002519157723820939245295475572, −8.733630903561318471582211222656, −8.247827016681358338699106327510, −7.53632864653844024945923131798, −7.46301279752361470901499498374, −6.61600017498811391917873139111, −6.19321040018157050389604631782, −5.38712706105755021847422947230, −4.62285613804853643208241606946, −3.78995178946695161507197151620, −3.39651452170084757716747861999, −3.21551153951879944953165713886, −2.00516068697428410265218115685, −1.60688453692534518598007565808, 1.60688453692534518598007565808, 2.00516068697428410265218115685, 3.21551153951879944953165713886, 3.39651452170084757716747861999, 3.78995178946695161507197151620, 4.62285613804853643208241606946, 5.38712706105755021847422947230, 6.19321040018157050389604631782, 6.61600017498811391917873139111, 7.46301279752361470901499498374, 7.53632864653844024945923131798, 8.247827016681358338699106327510, 8.733630903561318471582211222656, 9.002519157723820939245295475572, 9.389652658008486485429618465354

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