Properties

Label 2-1334-1.1-c1-0-32
Degree $2$
Conductor $1334$
Sign $1$
Analytic cond. $10.6520$
Root an. cond. $3.26374$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2.05·3-s + 4-s + 3.66·5-s − 2.05·6-s + 4.21·7-s − 8-s + 1.23·9-s − 3.66·10-s − 3.07·11-s + 2.05·12-s + 3.04·13-s − 4.21·14-s + 7.53·15-s + 16-s − 6.40·17-s − 1.23·18-s + 3.39·19-s + 3.66·20-s + 8.67·21-s + 3.07·22-s − 23-s − 2.05·24-s + 8.40·25-s − 3.04·26-s − 3.63·27-s + 4.21·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.18·3-s + 0.5·4-s + 1.63·5-s − 0.840·6-s + 1.59·7-s − 0.353·8-s + 0.411·9-s − 1.15·10-s − 0.925·11-s + 0.593·12-s + 0.845·13-s − 1.12·14-s + 1.94·15-s + 0.250·16-s − 1.55·17-s − 0.290·18-s + 0.779·19-s + 0.818·20-s + 1.89·21-s + 0.654·22-s − 0.208·23-s − 0.420·24-s + 1.68·25-s − 0.597·26-s − 0.699·27-s + 0.796·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1334\)    =    \(2 \cdot 23 \cdot 29\)
Sign: $1$
Analytic conductor: \(10.6520\)
Root analytic conductor: \(3.26374\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1334,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.692765011\)
\(L(\frac12)\) \(\approx\) \(2.692765011\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 - T \)
good3 \( 1 - 2.05T + 3T^{2} \)
5 \( 1 - 3.66T + 5T^{2} \)
7 \( 1 - 4.21T + 7T^{2} \)
11 \( 1 + 3.07T + 11T^{2} \)
13 \( 1 - 3.04T + 13T^{2} \)
17 \( 1 + 6.40T + 17T^{2} \)
19 \( 1 - 3.39T + 19T^{2} \)
31 \( 1 + 1.07T + 31T^{2} \)
37 \( 1 + 0.533T + 37T^{2} \)
41 \( 1 + 2.53T + 41T^{2} \)
43 \( 1 - 7.18T + 43T^{2} \)
47 \( 1 + 7.95T + 47T^{2} \)
53 \( 1 + 3.19T + 53T^{2} \)
59 \( 1 + 8.47T + 59T^{2} \)
61 \( 1 + 11.7T + 61T^{2} \)
67 \( 1 + 5.27T + 67T^{2} \)
71 \( 1 - 3.30T + 71T^{2} \)
73 \( 1 + 6.15T + 73T^{2} \)
79 \( 1 + 9.10T + 79T^{2} \)
83 \( 1 - 13.6T + 83T^{2} \)
89 \( 1 - 3.79T + 89T^{2} \)
97 \( 1 - 18.0T + 97T^{2} \)
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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

−9.254102213276291831109507619816, −8.959635691093620089369674921780, −8.148693918844911453750084531201, −7.58231618108122537968963771479, −6.38058794786585620804954206741, −5.51496403373086113143247878558, −4.59625102915497028247356385856, −3.01137774704123708665246071223, −2.12401939565870333318809052285, −1.56709946558897551870475970795, 1.56709946558897551870475970795, 2.12401939565870333318809052285, 3.01137774704123708665246071223, 4.59625102915497028247356385856, 5.51496403373086113143247878558, 6.38058794786585620804954206741, 7.58231618108122537968963771479, 8.148693918844911453750084531201, 8.959635691093620089369674921780, 9.254102213276291831109507619816

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