Properties

Label 2-93058-1.1-c1-0-9
Degree $2$
Conductor $93058$
Sign $1$
Analytic cond. $743.071$
Root an. cond. $27.2593$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 4-s + 2·6-s − 7-s − 8-s + 9-s − 4·11-s − 2·12-s + 14-s + 16-s − 18-s − 6·19-s + 2·21-s + 4·22-s + 23-s + 2·24-s − 5·25-s + 4·27-s − 28-s − 10·29-s − 4·31-s − 32-s + 8·33-s + 36-s + 2·37-s + 6·38-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.577·12-s + 0.267·14-s + 1/4·16-s − 0.235·18-s − 1.37·19-s + 0.436·21-s + 0.852·22-s + 0.208·23-s + 0.408·24-s − 25-s + 0.769·27-s − 0.188·28-s − 1.85·29-s − 0.718·31-s − 0.176·32-s + 1.39·33-s + 1/6·36-s + 0.328·37-s + 0.973·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93058 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93058 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93058\)    =    \(2 \cdot 7 \cdot 17^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(743.071\)
Root analytic conductor: \(27.2593\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{93058} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 93058,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
7 \( 1 + T \)
17 \( 1 \)
23 \( 1 - T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.38121916548001, −13.76302066397990, −13.06631573285009, −12.74123868957825, −12.39860675433610, −11.74576208684360, −11.06062478559989, −10.93307766135519, −10.60132871684171, −9.837623818485061, −9.493367511060464, −8.895277777308906, −8.270046805767287, −7.764163017718037, −7.283593125400489, −6.715111391520209, −6.117705968004187, −5.586145135439228, −5.460392018560054, −4.497141820845082, −3.992703054242778, −3.193053257258693, −2.401827970025648, −1.998682332784546, −0.9897778870696586, 0, 0, 0.9897778870696586, 1.998682332784546, 2.401827970025648, 3.193053257258693, 3.992703054242778, 4.497141820845082, 5.460392018560054, 5.586145135439228, 6.117705968004187, 6.715111391520209, 7.283593125400489, 7.764163017718037, 8.270046805767287, 8.895277777308906, 9.493367511060464, 9.837623818485061, 10.60132871684171, 10.93307766135519, 11.06062478559989, 11.74576208684360, 12.39860675433610, 12.74123868957825, 13.06631573285009, 13.76302066397990, 14.38121916548001

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