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Properties

Label 2-93138-1.1-c1-0-12
Degree $2$
Conductor $93138$
Sign $1$
Analytic cond. $743.710$
Root an. cond. $27.2710$
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 + 5·11-s − 12-s − 4·13-s − 2·15-s + 16-s + 18-s + 2·20-s + 5·22-s + 2·23-s − 24-s − 25-s − 4·26-s − 27-s − 2·29-s − 2·30-s + 7·31-s + 32-s − 5·33-s + 36-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 + 1.50·11-s − 0.288·12-s − 1.10·13-s − 0.516·15-s + 1/4·16-s + 0.235·18-s + 0.447·20-s + 1.06·22-s + 0.417·23-s − 0.204·24-s − 1/5·25-s − 0.784·26-s − 0.192·27-s − 0.371·29-s − 0.365·30-s + 1.25·31-s + 0.176·32-s − 0.870·33-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93138\)    =    \(2 \cdot 3 \cdot 19^{2} \cdot 43\)
Sign: $1$
Analytic conductor: \(743.710\)
Root analytic conductor: \(27.2710\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 93138,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.650174160\)
\(L(\frac12)\) \(\approx\) \(4.650174160\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
19 \( 1 \)
43 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−13.91089932478845, −13.26459689848567, −12.88080201781756, −12.40222298460664, −11.85334725954724, −11.37958539425354, −11.23625945739587, −10.21379657978475, −9.868600614723353, −9.651803588052857, −8.921680353405110, −8.310737865787937, −7.598401921029406, −6.982499944627185, −6.551000556402509, −6.242753502549432, −5.466079580537575, −5.250302388708596, −4.359589865446550, −4.199467068946622, −3.297033232453067, −2.646145610702072, −1.986018053005772, −1.399170823441039, −0.6376362473622169, 0.6376362473622169, 1.399170823441039, 1.986018053005772, 2.646145610702072, 3.297033232453067, 4.199467068946622, 4.359589865446550, 5.250302388708596, 5.466079580537575, 6.242753502549432, 6.551000556402509, 6.982499944627185, 7.598401921029406, 8.310737865787937, 8.921680353405110, 9.651803588052857, 9.868600614723353, 10.21379657978475, 11.23625945739587, 11.37958539425354, 11.85334725954724, 12.40222298460664, 12.88080201781756, 13.26459689848567, 13.91089932478845

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