Properties

Label 2-8025-1.1-c1-0-243
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  + 1.61·2-s − 3-s + 0.618·4-s − 1.61·6-s − 1.23·7-s − 2.23·8-s + 9-s − 2·11-s − 0.618·12-s + 13-s − 2.00·14-s − 4.85·16-s + 7.47·17-s + 1.61·18-s − 2.23·19-s + 1.23·21-s − 3.23·22-s + 4·23-s + 2.23·24-s + 1.61·26-s − 27-s − 0.763·28-s + 1.23·29-s − 2·31-s − 3.38·32-s + 2·33-s + 12.0·34-s + ⋯
L(s)  = 1  + 1.14·2-s − 0.577·3-s + 0.309·4-s − 0.660·6-s − 0.467·7-s − 0.790·8-s + 0.333·9-s − 0.603·11-s − 0.178·12-s + 0.277·13-s − 0.534·14-s − 1.21·16-s + 1.81·17-s + 0.381·18-s − 0.512·19-s + 0.269·21-s − 0.689·22-s + 0.834·23-s + 0.456·24-s + 0.317·26-s − 0.192·27-s − 0.144·28-s + 0.229·29-s − 0.359·31-s − 0.597·32-s + 0.348·33-s + 2.07·34-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 \( 1 - 1.61T + 2T^{2} \)
7 \( 1 + 1.23T + 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 - 7.47T + 17T^{2} \)
19 \( 1 + 2.23T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 1.23T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 - 7.94T + 37T^{2} \)
41 \( 1 + 7.23T + 41T^{2} \)
43 \( 1 + 4.47T + 43T^{2} \)
47 \( 1 + 7.70T + 47T^{2} \)
53 \( 1 - 3.52T + 53T^{2} \)
59 \( 1 - 8.94T + 59T^{2} \)
61 \( 1 - 7.94T + 61T^{2} \)
67 \( 1 + 1.52T + 67T^{2} \)
71 \( 1 + 8.70T + 71T^{2} \)
73 \( 1 - 3.70T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 13.7T + 83T^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−7.23364478070541773066492035082, −6.53358138018195434591214884713, −5.93019035014758498409729296383, −5.28981632039806019278882603763, −4.86665660378284655692390599575, −3.90534058451608448728005536771, −3.31157694363779819151537939095, −2.59807439228722807123824730793, −1.23800182648834983810266662841, 0, 1.23800182648834983810266662841, 2.59807439228722807123824730793, 3.31157694363779819151537939095, 3.90534058451608448728005536771, 4.86665660378284655692390599575, 5.28981632039806019278882603763, 5.93019035014758498409729296383, 6.53358138018195434591214884713, 7.23364478070541773066492035082

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