Properties

Label 2-4334-1.1-c1-0-13
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 − 2.03·3-s + 4-s + 3.12·5-s + 2.03·6-s − 4.32·7-s − 8-s + 1.13·9-s − 3.12·10-s − 11-s − 2.03·12-s − 2.76·13-s + 4.32·14-s − 6.36·15-s + 16-s + 0.311·17-s − 1.13·18-s + 2.52·19-s + 3.12·20-s + 8.78·21-s + 22-s − 4.26·23-s + 2.03·24-s + 4.78·25-s + 2.76·26-s + 3.78·27-s − 4.32·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.17·3-s + 0.5·4-s + 1.39·5-s + 0.830·6-s − 1.63·7-s − 0.353·8-s + 0.379·9-s − 0.989·10-s − 0.301·11-s − 0.587·12-s − 0.765·13-s + 1.15·14-s − 1.64·15-s + 0.250·16-s + 0.0755·17-s − 0.268·18-s + 0.579·19-s + 0.699·20-s + 1.91·21-s + 0.213·22-s − 0.889·23-s + 0.415·24-s + 0.957·25-s + 0.541·26-s + 0.728·27-s − 0.816·28-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

\(L(1)\) \(\approx\) \(0.4641777802\)
\(L(\frac12)\) \(\approx\) \(0.4641777802\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.03T + 3T^{2} \)
5 \( 1 - 3.12T + 5T^{2} \)
7 \( 1 + 4.32T + 7T^{2} \)
13 \( 1 + 2.76T + 13T^{2} \)
17 \( 1 - 0.311T + 17T^{2} \)
19 \( 1 - 2.52T + 19T^{2} \)
23 \( 1 + 4.26T + 23T^{2} \)
29 \( 1 - 1.03T + 29T^{2} \)
31 \( 1 + 9.13T + 31T^{2} \)
37 \( 1 + 10.4T + 37T^{2} \)
41 \( 1 - 4.95T + 41T^{2} \)
43 \( 1 + 5.49T + 43T^{2} \)
47 \( 1 + 5.87T + 47T^{2} \)
53 \( 1 - 6.48T + 53T^{2} \)
59 \( 1 - 9.98T + 59T^{2} \)
61 \( 1 - 7.19T + 61T^{2} \)
67 \( 1 + 8.12T + 67T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 - 5.38T + 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 - 8.19T + 83T^{2} \)
89 \( 1 + 7.18T + 89T^{2} \)
97 \( 1 + 9.69T + 97T^{2} \)
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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

−8.611942033754916281872145163864, −7.36109115640956864493467796255, −6.79632688921188472043388127128, −6.19135388446302090125970842284, −5.59496461253194744731089321364, −5.14200531877744665713992056437, −3.62871246508135529630474461118, −2.69808305853803020260734727678, −1.80788617679971419508678803504, −0.43280186104744185773414332008, 0.43280186104744185773414332008, 1.80788617679971419508678803504, 2.69808305853803020260734727678, 3.62871246508135529630474461118, 5.14200531877744665713992056437, 5.59496461253194744731089321364, 6.19135388446302090125970842284, 6.79632688921188472043388127128, 7.36109115640956864493467796255, 8.611942033754916281872145163864

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