Properties

Label 2-8025-1.1-c1-0-176
Degree $2$
Conductor $8025$
Sign $-1$
Analytic cond. $64.0799$
Root an. cond. $8.00499$
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  − 1.91·2-s + 3-s + 1.68·4-s − 1.91·6-s − 2.92·7-s + 0.612·8-s + 9-s − 4.17·11-s + 1.68·12-s + 3.33·13-s + 5.60·14-s − 4.53·16-s + 4.76·17-s − 1.91·18-s − 7.93·19-s − 2.92·21-s + 8.00·22-s − 8.94·23-s + 0.612·24-s − 6.39·26-s + 27-s − 4.91·28-s + 4.21·29-s + 5.71·31-s + 7.47·32-s − 4.17·33-s − 9.14·34-s + ⋯
L(s)  = 1  − 1.35·2-s + 0.577·3-s + 0.840·4-s − 0.783·6-s − 1.10·7-s + 0.216·8-s + 0.333·9-s − 1.25·11-s + 0.485·12-s + 0.925·13-s + 1.49·14-s − 1.13·16-s + 1.15·17-s − 0.452·18-s − 1.82·19-s − 0.637·21-s + 1.70·22-s − 1.86·23-s + 0.124·24-s − 1.25·26-s + 0.192·27-s − 0.928·28-s + 0.783·29-s + 1.02·31-s + 1.32·32-s − 0.726·33-s − 1.56·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8025\)    =    \(3 \cdot 5^{2} \cdot 107\)
Sign: $-1$
Analytic conductor: \(64.0799\)
Root analytic conductor: \(8.00499\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8025,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 - T \)
5 \( 1 \)
107 \( 1 - T \)
good2 \( 1 + 1.91T + 2T^{2} \)
7 \( 1 + 2.92T + 7T^{2} \)
11 \( 1 + 4.17T + 11T^{2} \)
13 \( 1 - 3.33T + 13T^{2} \)
17 \( 1 - 4.76T + 17T^{2} \)
19 \( 1 + 7.93T + 19T^{2} \)
23 \( 1 + 8.94T + 23T^{2} \)
29 \( 1 - 4.21T + 29T^{2} \)
31 \( 1 - 5.71T + 31T^{2} \)
37 \( 1 - 12.0T + 37T^{2} \)
41 \( 1 + 4.24T + 41T^{2} \)
43 \( 1 + 4.47T + 43T^{2} \)
47 \( 1 - 12.9T + 47T^{2} \)
53 \( 1 - 7.42T + 53T^{2} \)
59 \( 1 + 0.715T + 59T^{2} \)
61 \( 1 - 11.5T + 61T^{2} \)
67 \( 1 + 1.78T + 67T^{2} \)
71 \( 1 - 2.33T + 71T^{2} \)
73 \( 1 + 2.56T + 73T^{2} \)
79 \( 1 + 7.27T + 79T^{2} \)
83 \( 1 - 3.88T + 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + 6.04T + 97T^{2} \)
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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

−7.86568163577014253279094603651, −6.99833402389941743907981953183, −6.29304962508294098565644294222, −5.71093953935341942436274450675, −4.42821467278907832516680861465, −3.83380114629097090334544307562, −2.75189727406174018624096065399, −2.22437168569739361637809711403, −1.01680945628030937630681231690, 0, 1.01680945628030937630681231690, 2.22437168569739361637809711403, 2.75189727406174018624096065399, 3.83380114629097090334544307562, 4.42821467278907832516680861465, 5.71093953935341942436274450675, 6.29304962508294098565644294222, 6.99833402389941743907981953183, 7.86568163577014253279094603651

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