Properties

Label 2-91e2-1.1-c1-0-364
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.656·2-s − 0.204·3-s − 1.56·4-s + 1.35·5-s − 0.134·6-s − 2.34·8-s − 2.95·9-s + 0.892·10-s + 1.90·11-s + 0.321·12-s − 0.278·15-s + 1.60·16-s + 3.56·17-s − 1.94·18-s − 0.985·19-s − 2.13·20-s + 1.25·22-s + 1.69·23-s + 0.479·24-s − 3.15·25-s + 1.21·27-s + 6.54·29-s − 0.182·30-s − 7.69·31-s + 5.73·32-s − 0.390·33-s + 2.34·34-s + ⋯
L(s)  = 1  + 0.463·2-s − 0.118·3-s − 0.784·4-s + 0.608·5-s − 0.0548·6-s − 0.828·8-s − 0.986·9-s + 0.282·10-s + 0.574·11-s + 0.0927·12-s − 0.0718·15-s + 0.400·16-s + 0.864·17-s − 0.457·18-s − 0.226·19-s − 0.477·20-s + 0.266·22-s + 0.353·23-s + 0.0978·24-s − 0.630·25-s + 0.234·27-s + 1.21·29-s − 0.0333·30-s − 1.38·31-s + 1.01·32-s − 0.0679·33-s + 0.401·34-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: \(1\)
Selberg data: \((2,\ 8281,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.656T + 2T^{2} \)
3 \( 1 + 0.204T + 3T^{2} \)
5 \( 1 - 1.35T + 5T^{2} \)
11 \( 1 - 1.90T + 11T^{2} \)
17 \( 1 - 3.56T + 17T^{2} \)
19 \( 1 + 0.985T + 19T^{2} \)
23 \( 1 - 1.69T + 23T^{2} \)
29 \( 1 - 6.54T + 29T^{2} \)
31 \( 1 + 7.69T + 31T^{2} \)
37 \( 1 - 2.02T + 37T^{2} \)
41 \( 1 + 9.88T + 41T^{2} \)
43 \( 1 + 3.16T + 43T^{2} \)
47 \( 1 + 7.76T + 47T^{2} \)
53 \( 1 - 0.354T + 53T^{2} \)
59 \( 1 + 2.16T + 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 - 9.05T + 71T^{2} \)
73 \( 1 - 7.13T + 73T^{2} \)
79 \( 1 + 5.39T + 79T^{2} \)
83 \( 1 - 2.03T + 83T^{2} \)
89 \( 1 - 6.89T + 89T^{2} \)
97 \( 1 - 14.6T + 97T^{2} \)
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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

−7.47471045594710939824721421469, −6.40692748606181296530646447718, −6.08317807152053092865562182046, −5.19065346780208609813190192519, −4.94660266588747611222586140547, −3.74938636099151667818091477826, −3.33537730490894149019626877718, −2.33577796029993186202790757595, −1.21399960076514160510021024523, 0, 1.21399960076514160510021024523, 2.33577796029993186202790757595, 3.33537730490894149019626877718, 3.74938636099151667818091477826, 4.94660266588747611222586140547, 5.19065346780208609813190192519, 6.08317807152053092865562182046, 6.40692748606181296530646447718, 7.47471045594710939824721421469

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