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Properties

Label 4-99e2-1.1-c1e2-0-2
Degree $4$
Conductor $9801$
Sign $1$
Analytic cond. $0.624920$
Root an. cond. $0.889111$
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 + 2·4-s + 2·5-s − 4·7-s + 4·8-s − 3·9-s + 4·10-s − 11-s − 4·13-s − 8·14-s + 8·16-s + 8·17-s − 6·18-s − 12·19-s + 4·20-s − 2·22-s + 23-s + 5·25-s − 8·26-s − 8·28-s − 31-s + 8·32-s + 16·34-s − 8·35-s − 6·36-s + 6·37-s − 24·38-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 0.894·5-s − 1.51·7-s + 1.41·8-s − 9-s + 1.26·10-s − 0.301·11-s − 1.10·13-s − 2.13·14-s + 2·16-s + 1.94·17-s − 1.41·18-s − 2.75·19-s + 0.894·20-s − 0.426·22-s + 0.208·23-s + 25-s − 1.56·26-s − 1.51·28-s − 0.179·31-s + 1.41·32-s + 2.74·34-s − 1.35·35-s − 36-s + 0.986·37-s − 3.89·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9801\)    =    \(3^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(0.624920\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9801,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.878160901\)
\(L(\frac12)\) \(\approx\) \(1.878160901\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
11$C_2$ \( 1 + T + T^{2} \)
good2$C_2^2$ \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
13$C_2^2$ \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 12 T + 101 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 7 T + 2 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 11 T + 62 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
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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

−14.23418899392058021219652899344, −13.58480207612691748941818361235, −13.09469991185346332045386583260, −12.62844252177810186401158605939, −12.53130220306551015781220229495, −11.85230104161715369730063398338, −10.74931166116132776070400334915, −10.66881913959611422418747613477, −9.834799830741763466408613400424, −9.653262976223268069337490628147, −8.639877389633739775295062926979, −8.017926436543149103413944378313, −7.19537421954810871040449130182, −6.49567444727210546409205802887, −6.00774878416193081244758170367, −5.41240376141583273152091738853, −4.81185525673423608439381107896, −3.87777778052902744490531742535, −3.06110036737768672608680415369, −2.29631348731773603655789539034, 2.29631348731773603655789539034, 3.06110036737768672608680415369, 3.87777778052902744490531742535, 4.81185525673423608439381107896, 5.41240376141583273152091738853, 6.00774878416193081244758170367, 6.49567444727210546409205802887, 7.19537421954810871040449130182, 8.017926436543149103413944378313, 8.639877389633739775295062926979, 9.653262976223268069337490628147, 9.834799830741763466408613400424, 10.66881913959611422418747613477, 10.74931166116132776070400334915, 11.85230104161715369730063398338, 12.53130220306551015781220229495, 12.62844252177810186401158605939, 13.09469991185346332045386583260, 13.58480207612691748941818361235, 14.23418899392058021219652899344

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