Properties

Label 2-975-1.1-c1-0-18
Degree $2$
Conductor $975$
Sign $-1$
Analytic cond. $7.78541$
Root an. cond. $2.79023$
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  − 2.51·2-s − 3-s + 4.32·4-s + 2.51·6-s + 0.514·7-s − 5.83·8-s + 9-s − 0.806·11-s − 4.32·12-s − 13-s − 1.29·14-s + 6.02·16-s + 2.02·17-s − 2.51·18-s + 0.292·19-s − 0.514·21-s + 2.02·22-s − 8.34·23-s + 5.83·24-s + 2.51·26-s − 27-s + 2.22·28-s − 9.64·29-s + 6.22·31-s − 3.48·32-s + 0.806·33-s − 5.09·34-s + ⋯
L(s)  = 1  − 1.77·2-s − 0.577·3-s + 2.16·4-s + 1.02·6-s + 0.194·7-s − 2.06·8-s + 0.333·9-s − 0.243·11-s − 1.24·12-s − 0.277·13-s − 0.345·14-s + 1.50·16-s + 0.491·17-s − 0.592·18-s + 0.0671·19-s − 0.112·21-s + 0.432·22-s − 1.74·23-s + 1.19·24-s + 0.493·26-s − 0.192·27-s + 0.419·28-s − 1.79·29-s + 1.11·31-s − 0.616·32-s + 0.140·33-s − 0.874·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -1)\)

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 \( 1 \)
13 \( 1 + T \)
good2 \( 1 + 2.51T + 2T^{2} \)
7 \( 1 - 0.514T + 7T^{2} \)
11 \( 1 + 0.806T + 11T^{2} \)
17 \( 1 - 2.02T + 17T^{2} \)
19 \( 1 - 0.292T + 19T^{2} \)
23 \( 1 + 8.34T + 23T^{2} \)
29 \( 1 + 9.64T + 29T^{2} \)
31 \( 1 - 6.22T + 31T^{2} \)
37 \( 1 - 6.34T + 37T^{2} \)
41 \( 1 + 1.70T + 41T^{2} \)
43 \( 1 - 7.96T + 43T^{2} \)
47 \( 1 - 10.7T + 47T^{2} \)
53 \( 1 + 4.70T + 53T^{2} \)
59 \( 1 - 4.12T + 59T^{2} \)
61 \( 1 + 2.61T + 61T^{2} \)
67 \( 1 + 13.5T + 67T^{2} \)
71 \( 1 + 7.70T + 71T^{2} \)
73 \( 1 + 11.3T + 73T^{2} \)
79 \( 1 + 4.73T + 79T^{2} \)
83 \( 1 - 6.57T + 83T^{2} \)
89 \( 1 + 0.971T + 89T^{2} \)
97 \( 1 + 9.61T + 97T^{2} \)
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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

−9.673184490035068719851929271002, −8.866362479062510072994527433662, −7.78007775450077151448623043303, −7.54711719853155179812786791618, −6.34201168444215658603698384617, −5.64541376039925194511285020555, −4.20288599748644012999333815357, −2.58615382366371990052800286217, −1.43048443842338719105823122216, 0, 1.43048443842338719105823122216, 2.58615382366371990052800286217, 4.20288599748644012999333815357, 5.64541376039925194511285020555, 6.34201168444215658603698384617, 7.54711719853155179812786791618, 7.78007775450077151448623043303, 8.866362479062510072994527433662, 9.673184490035068719851929271002

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