Properties

Label 4-176868-1.1-c1e2-0-3
Degree $4$
Conductor $176868$
Sign $1$
Analytic cond. $11.2772$
Root an. cond. $1.83252$
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  + 2·2-s + 3·4-s + 4·8-s + 9-s − 4·13-s + 5·16-s + 17-s + 2·18-s + 8·19-s − 6·25-s − 8·26-s + 6·32-s + 2·34-s + 3·36-s + 16·38-s + 24·43-s − 14·49-s − 12·50-s − 12·52-s + 12·53-s + 24·59-s + 7·64-s − 24·67-s + 3·68-s + 4·72-s + 24·76-s + 81-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.41·8-s + 1/3·9-s − 1.10·13-s + 5/4·16-s + 0.242·17-s + 0.471·18-s + 1.83·19-s − 6/5·25-s − 1.56·26-s + 1.06·32-s + 0.342·34-s + 1/2·36-s + 2.59·38-s + 3.65·43-s − 2·49-s − 1.69·50-s − 1.66·52-s + 1.64·53-s + 3.12·59-s + 7/8·64-s − 2.93·67-s + 0.363·68-s + 0.471·72-s + 2.75·76-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176868 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176868 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(176868\)    =    \(2^{2} \cdot 3^{2} \cdot 17^{3}\)
Sign: $1$
Analytic conductor: \(11.2772\)
Root analytic conductor: \(1.83252\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{176868} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 176868,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.248150726\)
\(L(\frac12)\) \(\approx\) \(4.248150726\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
17$C_1$ \( 1 - T \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−9.547040289033150690294822441027, −8.613628586363707276957269542969, −7.958882066985845549410362974675, −7.37401646943555492580120617253, −7.36440720073199929079936076774, −6.73915837939670470281220727226, −5.97447010434826597680072551041, −5.51457882926074302977186886250, −5.32734843201609780311424178486, −4.47367404653766349933628761316, −4.13875090005906653574798019221, −3.49143456848613430686623125279, −2.76309813409348720224135008444, −2.28380910146500603557363711542, −1.19155942635042385588089355891, 1.19155942635042385588089355891, 2.28380910146500603557363711542, 2.76309813409348720224135008444, 3.49143456848613430686623125279, 4.13875090005906653574798019221, 4.47367404653766349933628761316, 5.32734843201609780311424178486, 5.51457882926074302977186886250, 5.97447010434826597680072551041, 6.73915837939670470281220727226, 7.36440720073199929079936076774, 7.37401646943555492580120617253, 7.958882066985845549410362974675, 8.613628586363707276957269542969, 9.547040289033150690294822441027

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