Properties

Label 4-107811-1.1-c1e2-0-0
Degree $4$
Conductor $107811$
Sign $1$
Analytic cond. $6.87412$
Root an. cond. $1.61921$
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 − 4-s − 8·8-s + 11-s − 7·16-s − 4·17-s + 2·22-s + 6·25-s + 12·29-s + 8·31-s + 14·32-s − 8·34-s − 12·37-s + 20·41-s − 44-s − 10·49-s + 12·50-s + 24·58-s + 16·62-s + 35·64-s + 16·67-s + 4·68-s − 24·74-s + 40·82-s − 24·83-s − 8·88-s + 4·97-s + ⋯
L(s)  = 1  + 1.41·2-s − 1/2·4-s − 2.82·8-s + 0.301·11-s − 7/4·16-s − 0.970·17-s + 0.426·22-s + 6/5·25-s + 2.22·29-s + 1.43·31-s + 2.47·32-s − 1.37·34-s − 1.97·37-s + 3.12·41-s − 0.150·44-s − 1.42·49-s + 1.69·50-s + 3.15·58-s + 2.03·62-s + 35/8·64-s + 1.95·67-s + 0.485·68-s − 2.78·74-s + 4.41·82-s − 2.63·83-s − 0.852·88-s + 0.406·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107811 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107811 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(107811\)    =    \(3^{4} \cdot 11^{3}\)
Sign: $1$
Analytic conductor: \(6.87412\)
Root analytic conductor: \(1.61921\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{107811} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 107811,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.944057623\)
\(L(\frac12)\) \(\approx\) \(1.944057623\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
good2$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
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

−9.602153864880356635903308889209, −8.765403563089383948261377682178, −8.702375538631797576207794069914, −8.310039341701636953836538691364, −7.45345434128145669094314403830, −6.52495869918911105195591443455, −6.51037497293636656959710624288, −5.82642873508535905813523337142, −5.14146744035381483907036545967, −4.65639191028282856260150309872, −4.44676656528528856043784213256, −3.75599253613222283098497330601, −3.03530616104555310081477111018, −2.56141659749378916924669768053, −0.843916200143676929117242957092, 0.843916200143676929117242957092, 2.56141659749378916924669768053, 3.03530616104555310081477111018, 3.75599253613222283098497330601, 4.44676656528528856043784213256, 4.65639191028282856260150309872, 5.14146744035381483907036545967, 5.82642873508535905813523337142, 6.51037497293636656959710624288, 6.52495869918911105195591443455, 7.45345434128145669094314403830, 8.310039341701636953836538691364, 8.702375538631797576207794069914, 8.765403563089383948261377682178, 9.602153864880356635903308889209

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