Properties

Label 2-209814-1.1-c1-0-6
Degree $2$
Conductor $209814$
Sign $1$
Analytic cond. $1675.37$
Root an. cond. $40.9313$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 2·5-s + 6-s − 8-s + 9-s − 2·10-s − 12-s + 2·13-s − 2·15-s + 16-s − 18-s − 4·19-s + 2·20-s + 24-s − 25-s − 2·26-s − 27-s − 10·29-s + 2·30-s − 8·31-s − 32-s + 36-s + 2·37-s + 4·38-s − 2·39-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.288·12-s + 0.554·13-s − 0.516·15-s + 1/4·16-s − 0.235·18-s − 0.917·19-s + 0.447·20-s + 0.204·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s − 1.85·29-s + 0.365·30-s − 1.43·31-s − 0.176·32-s + 1/6·36-s + 0.328·37-s + 0.648·38-s − 0.320·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209814 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209814 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(209814\)    =    \(2 \cdot 3 \cdot 11^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(1675.37\)
Root analytic conductor: \(40.9313\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 209814,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5812047095\)
\(L(\frac12)\) \(\approx\) \(0.5812047095\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
11 \( 1 \)
17 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.04692397817100, −12.67056541084669, −11.92037757932040, −11.46800653910542, −11.09147213900404, −10.63897328501242, −10.27331749715832, −9.532650472038098, −9.498198523780816, −8.803892962334295, −8.429434660790663, −7.666645740517029, −7.382098903943955, −6.723939362760841, −6.247924841075336, −5.842628703366251, −5.457888889292154, −4.876616501677762, −4.030963345070786, −3.710675047313068, −2.885944476613487, −2.133445133830167, −1.766963879451072, −1.232211170160349, −0.2473208441737513, 0.2473208441737513, 1.232211170160349, 1.766963879451072, 2.133445133830167, 2.885944476613487, 3.710675047313068, 4.030963345070786, 4.876616501677762, 5.457888889292154, 5.842628703366251, 6.247924841075336, 6.723939362760841, 7.382098903943955, 7.666645740517029, 8.429434660790663, 8.803892962334295, 9.498198523780816, 9.532650472038098, 10.27331749715832, 10.63897328501242, 11.09147213900404, 11.46800653910542, 11.92037757932040, 12.67056541084669, 13.04692397817100

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