Properties

Label 2-4334-1.1-c1-0-81
Degree $2$
Conductor $4334$
Sign $1$
Analytic cond. $34.6071$
Root an. cond. $5.88278$
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 + 2.93·3-s + 4-s − 2.47·5-s − 2.93·6-s + 4.98·7-s − 8-s + 5.59·9-s + 2.47·10-s − 11-s + 2.93·12-s + 4.23·13-s − 4.98·14-s − 7.26·15-s + 16-s + 0.103·17-s − 5.59·18-s − 1.47·19-s − 2.47·20-s + 14.6·21-s + 22-s + 7.68·23-s − 2.93·24-s + 1.14·25-s − 4.23·26-s + 7.61·27-s + 4.98·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.69·3-s + 0.5·4-s − 1.10·5-s − 1.19·6-s + 1.88·7-s − 0.353·8-s + 1.86·9-s + 0.783·10-s − 0.301·11-s + 0.846·12-s + 1.17·13-s − 1.33·14-s − 1.87·15-s + 0.250·16-s + 0.0251·17-s − 1.31·18-s − 0.338·19-s − 0.554·20-s + 3.18·21-s + 0.213·22-s + 1.60·23-s − 0.598·24-s + 0.229·25-s − 0.829·26-s + 1.46·27-s + 0.941·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4334\)    =    \(2 \cdot 11 \cdot 197\)
Sign: $1$
Analytic conductor: \(34.6071\)
Root analytic conductor: \(5.88278\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4334,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.996718674\)
\(L(\frac12)\) \(\approx\) \(2.996718674\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
11 \( 1 + T \)
197 \( 1 - T \)
good3 \( 1 - 2.93T + 3T^{2} \)
5 \( 1 + 2.47T + 5T^{2} \)
7 \( 1 - 4.98T + 7T^{2} \)
13 \( 1 - 4.23T + 13T^{2} \)
17 \( 1 - 0.103T + 17T^{2} \)
19 \( 1 + 1.47T + 19T^{2} \)
23 \( 1 - 7.68T + 23T^{2} \)
29 \( 1 - 2.31T + 29T^{2} \)
31 \( 1 - 5.41T + 31T^{2} \)
37 \( 1 + 1.88T + 37T^{2} \)
41 \( 1 + 6.79T + 41T^{2} \)
43 \( 1 + 5.78T + 43T^{2} \)
47 \( 1 + 3.21T + 47T^{2} \)
53 \( 1 - 1.01T + 53T^{2} \)
59 \( 1 + 3.45T + 59T^{2} \)
61 \( 1 - 2.90T + 61T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 - 0.898T + 71T^{2} \)
73 \( 1 - 13.7T + 73T^{2} \)
79 \( 1 + 0.156T + 79T^{2} \)
83 \( 1 - 5.32T + 83T^{2} \)
89 \( 1 - 9.25T + 89T^{2} \)
97 \( 1 - 9.71T + 97T^{2} \)
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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

−8.294656763058153889951509481181, −8.056688058841910931336511144323, −7.37267774251831505912681247239, −6.63711994379289692873898098478, −5.15919785212745920876259175916, −4.42760190245964517375480531786, −3.62959124371499339378586402772, −2.87690191314510953157544058773, −1.85675116237902164338009561166, −1.10896158832323310043992321974, 1.10896158832323310043992321974, 1.85675116237902164338009561166, 2.87690191314510953157544058773, 3.62959124371499339378586402772, 4.42760190245964517375480531786, 5.15919785212745920876259175916, 6.63711994379289692873898098478, 7.37267774251831505912681247239, 8.056688058841910931336511144323, 8.294656763058153889951509481181

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