Properties

Label 2-91e2-1.1-c1-0-252
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  + 1.21·2-s + 1.74·3-s − 0.534·4-s + 2.21·5-s + 2.11·6-s − 3.06·8-s + 0.0444·9-s + 2.67·10-s + 0.789·11-s − 0.931·12-s + 3.85·15-s − 2.64·16-s + 1.74·17-s + 0.0537·18-s + 4.32·19-s − 1.18·20-s + 0.955·22-s − 1.11·23-s − 5.35·24-s − 0.112·25-s − 5.15·27-s − 8.48·29-s + 4.67·30-s + 5.70·31-s + 2.93·32-s + 1.37·33-s + 2.11·34-s + ⋯
L(s)  = 1  + 0.856·2-s + 1.00·3-s − 0.267·4-s + 0.988·5-s + 0.862·6-s − 1.08·8-s + 0.0148·9-s + 0.846·10-s + 0.237·11-s − 0.269·12-s + 0.995·15-s − 0.661·16-s + 0.423·17-s + 0.0126·18-s + 0.991·19-s − 0.264·20-s + 0.203·22-s − 0.231·23-s − 1.09·24-s − 0.0225·25-s − 0.992·27-s − 1.57·29-s + 0.852·30-s + 1.02·31-s + 0.518·32-s + 0.239·33-s + 0.362·34-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: $\chi_{8281} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8281,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.785023391\)
\(L(\frac12)\) \(\approx\) \(4.785023391\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.21T + 2T^{2} \)
3 \( 1 - 1.74T + 3T^{2} \)
5 \( 1 - 2.21T + 5T^{2} \)
11 \( 1 - 0.789T + 11T^{2} \)
17 \( 1 - 1.74T + 17T^{2} \)
19 \( 1 - 4.32T + 19T^{2} \)
23 \( 1 + 1.11T + 23T^{2} \)
29 \( 1 + 8.48T + 29T^{2} \)
31 \( 1 - 5.70T + 31T^{2} \)
37 \( 1 + 2.27T + 37T^{2} \)
41 \( 1 - 12.1T + 41T^{2} \)
43 \( 1 - 8.06T + 43T^{2} \)
47 \( 1 - 8.74T + 47T^{2} \)
53 \( 1 - 7.95T + 53T^{2} \)
59 \( 1 - 10.9T + 59T^{2} \)
61 \( 1 - 13.0T + 61T^{2} \)
67 \( 1 + 6.55T + 67T^{2} \)
71 \( 1 + 5.85T + 71T^{2} \)
73 \( 1 - 8.00T + 73T^{2} \)
79 \( 1 + 6.91T + 79T^{2} \)
83 \( 1 - 3.14T + 83T^{2} \)
89 \( 1 - 3.39T + 89T^{2} \)
97 \( 1 + 0.0981T + 97T^{2} \)
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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

−7.74115735386912725160584352597, −7.21159741375855261629061502910, −5.94715506852979863448943507247, −5.82916513184496964069658724978, −5.08425461536798465618062388197, −4.02542808054566654085491915691, −3.63825766967085331467122271132, −2.66466018407331687136427332055, −2.22528494464595079400037718683, −0.908673617798597382957446926800, 0.908673617798597382957446926800, 2.22528494464595079400037718683, 2.66466018407331687136427332055, 3.63825766967085331467122271132, 4.02542808054566654085491915691, 5.08425461536798465618062388197, 5.82916513184496964069658724978, 5.94715506852979863448943507247, 7.21159741375855261629061502910, 7.74115735386912725160584352597

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