Properties

Label 2-91e2-1.1-c1-0-139
Degree $2$
Conductor $8281$
Sign $1$
Analytic cond. $66.1241$
Root an. cond. $8.13167$
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  − 2.24·2-s + 0.554·3-s + 3.04·4-s + 1.44·5-s − 1.24·6-s − 2.35·8-s − 2.69·9-s − 3.24·10-s − 2.55·11-s + 1.69·12-s + 0.801·15-s − 0.801·16-s + 5.29·17-s + 6.04·18-s + 5.85·19-s + 4.40·20-s + 5.74·22-s − 1.89·23-s − 1.30·24-s − 2.91·25-s − 3.15·27-s + 2.26·29-s − 1.80·30-s + 4.26·31-s + 6.51·32-s − 1.41·33-s − 11.8·34-s + ⋯
L(s)  = 1  − 1.58·2-s + 0.320·3-s + 1.52·4-s + 0.646·5-s − 0.509·6-s − 0.833·8-s − 0.897·9-s − 1.02·10-s − 0.770·11-s + 0.488·12-s + 0.207·15-s − 0.200·16-s + 1.28·17-s + 1.42·18-s + 1.34·19-s + 0.985·20-s + 1.22·22-s − 0.394·23-s − 0.266·24-s − 0.582·25-s − 0.607·27-s + 0.421·29-s − 0.328·30-s + 0.766·31-s + 1.15·32-s − 0.246·33-s − 2.04·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8281\)    =    \(7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(66.1241\)
Root analytic conductor: \(8.13167\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8281,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.019715749\)
\(L(\frac12)\) \(\approx\) \(1.019715749\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + 2.24T + 2T^{2} \)
3 \( 1 - 0.554T + 3T^{2} \)
5 \( 1 - 1.44T + 5T^{2} \)
11 \( 1 + 2.55T + 11T^{2} \)
17 \( 1 - 5.29T + 17T^{2} \)
19 \( 1 - 5.85T + 19T^{2} \)
23 \( 1 + 1.89T + 23T^{2} \)
29 \( 1 - 2.26T + 29T^{2} \)
31 \( 1 - 4.26T + 31T^{2} \)
37 \( 1 - 5.35T + 37T^{2} \)
41 \( 1 + 1.27T + 41T^{2} \)
43 \( 1 - 6.13T + 43T^{2} \)
47 \( 1 - 2.95T + 47T^{2} \)
53 \( 1 - 5.52T + 53T^{2} \)
59 \( 1 - 12.2T + 59T^{2} \)
61 \( 1 + 8.56T + 61T^{2} \)
67 \( 1 - 0.576T + 67T^{2} \)
71 \( 1 + 4.59T + 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 + 15.7T + 79T^{2} \)
83 \( 1 + 7.72T + 83T^{2} \)
89 \( 1 + 6.61T + 89T^{2} \)
97 \( 1 + 11.9T + 97T^{2} \)
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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

−7.994951001579711470592528797176, −7.48781342138460974854024901451, −6.67786507612454135747146899772, −5.62989170303597734415709693474, −5.51935990129035355869706274337, −4.17145705120195006988856874221, −2.94535556257234423824324230584, −2.57701297590639222403505283003, −1.52754232485701304798994006600, −0.64489129061817965521127828556, 0.64489129061817965521127828556, 1.52754232485701304798994006600, 2.57701297590639222403505283003, 2.94535556257234423824324230584, 4.17145705120195006988856874221, 5.51935990129035355869706274337, 5.62989170303597734415709693474, 6.67786507612454135747146899772, 7.48781342138460974854024901451, 7.994951001579711470592528797176

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