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Properties

Label 4-99e2-1.1-c1e2-0-1
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 2·4-s + 3·5-s + 4·7-s + 6·9-s − 11-s − 6·12-s − 2·13-s − 9·15-s − 12·17-s + 4·19-s + 6·20-s − 12·21-s − 3·23-s + 5·25-s − 9·27-s + 8·28-s + 6·29-s − 8·31-s + 3·33-s + 12·35-s + 12·36-s + 4·37-s + 6·39-s − 8·43-s − 2·44-s + 18·45-s + ⋯
L(s)  = 1  − 1.73·3-s + 4-s + 1.34·5-s + 1.51·7-s + 2·9-s − 0.301·11-s − 1.73·12-s − 0.554·13-s − 2.32·15-s − 2.91·17-s + 0.917·19-s + 1.34·20-s − 2.61·21-s − 0.625·23-s + 25-s − 1.73·27-s + 1.51·28-s + 1.11·29-s − 1.43·31-s + 0.522·33-s + 2.02·35-s + 2·36-s + 0.657·37-s + 0.960·39-s − 1.21·43-s − 0.301·44-s + 2.68·45-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

\(L(1)\) \(\approx\) \(0.9495262006\)
\(L(\frac12)\) \(\approx\) \(0.9495262006\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T + p T^{2} \)
11$C_2$ \( 1 + T + T^{2} \)
good2$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 2 T - 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 18 T + 241 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−13.92475801639020779272545550253, −13.68456216274581448742508932732, −13.03249938638593505731519680935, −12.50295702500184052377755686661, −11.67494008311005808829522411419, −11.59036606723586378882373788205, −10.94559451476732872807593857280, −10.76853697888891159864304747996, −10.14676737573860952850416345980, −9.454431631110923038638750997477, −8.799610798073646460970337728645, −7.901228967415635956236179770929, −7.14261781172656268895974449960, −6.59449577592498389311843276610, −6.27507846626751700891588224654, −5.37548419439968430942839636954, −4.98208456973095822275834559813, −4.38376850482268117782348579761, −2.33353216978153844795617437488, −1.78227959678888008814466134459, 1.78227959678888008814466134459, 2.33353216978153844795617437488, 4.38376850482268117782348579761, 4.98208456973095822275834559813, 5.37548419439968430942839636954, 6.27507846626751700891588224654, 6.59449577592498389311843276610, 7.14261781172656268895974449960, 7.901228967415635956236179770929, 8.799610798073646460970337728645, 9.454431631110923038638750997477, 10.14676737573860952850416345980, 10.76853697888891159864304747996, 10.94559451476732872807593857280, 11.59036606723586378882373788205, 11.67494008311005808829522411419, 12.50295702500184052377755686661, 13.03249938638593505731519680935, 13.68456216274581448742508932732, 13.92475801639020779272545550253

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