Properties

Label 2-8512-1.1-c1-0-25
Degree $2$
Conductor $8512$
Sign $1$
Analytic cond. $67.9686$
Root an. cond. $8.24431$
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  + 0.811·3-s − 0.636·5-s − 7-s − 2.34·9-s − 1.29·11-s − 1.02·13-s − 0.516·15-s − 7.09·17-s + 19-s − 0.811·21-s − 3.70·23-s − 4.59·25-s − 4.33·27-s − 4.94·29-s + 2.96·31-s − 1.05·33-s + 0.636·35-s − 2.78·37-s − 0.828·39-s + 5.64·41-s − 1.08·43-s + 1.48·45-s + 11.2·47-s + 49-s − 5.75·51-s + 6.23·53-s + 0.825·55-s + ⋯
L(s)  = 1  + 0.468·3-s − 0.284·5-s − 0.377·7-s − 0.780·9-s − 0.391·11-s − 0.283·13-s − 0.133·15-s − 1.72·17-s + 0.229·19-s − 0.177·21-s − 0.771·23-s − 0.919·25-s − 0.834·27-s − 0.918·29-s + 0.532·31-s − 0.183·33-s + 0.107·35-s − 0.457·37-s − 0.132·39-s + 0.881·41-s − 0.165·43-s + 0.221·45-s + 1.63·47-s + 0.142·49-s − 0.806·51-s + 0.857·53-s + 0.111·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8512\)    =    \(2^{6} \cdot 7 \cdot 19\)
Sign: $1$
Analytic conductor: \(67.9686\)
Root analytic conductor: \(8.24431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8512,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.033621924\)
\(L(\frac12)\) \(\approx\) \(1.033621924\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
19 \( 1 - T \)
good3 \( 1 - 0.811T + 3T^{2} \)
5 \( 1 + 0.636T + 5T^{2} \)
11 \( 1 + 1.29T + 11T^{2} \)
13 \( 1 + 1.02T + 13T^{2} \)
17 \( 1 + 7.09T + 17T^{2} \)
23 \( 1 + 3.70T + 23T^{2} \)
29 \( 1 + 4.94T + 29T^{2} \)
31 \( 1 - 2.96T + 31T^{2} \)
37 \( 1 + 2.78T + 37T^{2} \)
41 \( 1 - 5.64T + 41T^{2} \)
43 \( 1 + 1.08T + 43T^{2} \)
47 \( 1 - 11.2T + 47T^{2} \)
53 \( 1 - 6.23T + 53T^{2} \)
59 \( 1 + 1.27T + 59T^{2} \)
61 \( 1 - 11.3T + 61T^{2} \)
67 \( 1 + 7.11T + 67T^{2} \)
71 \( 1 - 4.45T + 71T^{2} \)
73 \( 1 + 2.12T + 73T^{2} \)
79 \( 1 - 13.1T + 79T^{2} \)
83 \( 1 + 2.08T + 83T^{2} \)
89 \( 1 - 7.92T + 89T^{2} \)
97 \( 1 - 9.66T + 97T^{2} \)
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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

−7.72961778662445013636968132176, −7.28824155661925908715011318098, −6.34073426624168775997540125686, −5.81139121391015738485781469393, −4.98463744833399596467434061899, −4.08362343646535925969837915930, −3.55130060072728447283973844501, −2.47727663257257431154858621512, −2.13789277956346894997400320308, −0.44959341818561389998374778731, 0.44959341818561389998374778731, 2.13789277956346894997400320308, 2.47727663257257431154858621512, 3.55130060072728447283973844501, 4.08362343646535925969837915930, 4.98463744833399596467434061899, 5.81139121391015738485781469393, 6.34073426624168775997540125686, 7.28824155661925908715011318098, 7.72961778662445013636968132176

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