Properties

Label 2-975-1.1-c1-0-1
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.414·2-s − 3-s − 1.82·4-s + 0.414·6-s − 2.82·7-s + 1.58·8-s + 9-s − 2·11-s + 1.82·12-s + 13-s + 1.17·14-s + 3·16-s − 7.65·17-s − 0.414·18-s − 2.82·19-s + 2.82·21-s + 0.828·22-s + 4·23-s − 1.58·24-s − 0.414·26-s − 27-s + 5.17·28-s + 2·29-s − 1.17·31-s − 4.41·32-s + 2·33-s + 3.17·34-s + ⋯
L(s)  = 1  − 0.292·2-s − 0.577·3-s − 0.914·4-s + 0.169·6-s − 1.06·7-s + 0.560·8-s + 0.333·9-s − 0.603·11-s + 0.527·12-s + 0.277·13-s + 0.313·14-s + 0.750·16-s − 1.85·17-s − 0.0976·18-s − 0.648·19-s + 0.617·21-s + 0.176·22-s + 0.834·23-s − 0.323·24-s − 0.0812·26-s − 0.192·27-s + 0.977·28-s + 0.371·29-s − 0.210·31-s − 0.780·32-s + 0.348·33-s + 0.543·34-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: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5394985515\)
\(L(\frac12)\) \(\approx\) \(0.5394985515\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.414T + 2T^{2} \)
7 \( 1 + 2.82T + 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
17 \( 1 + 7.65T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + 1.17T + 31T^{2} \)
37 \( 1 - 7.65T + 37T^{2} \)
41 \( 1 - 5.17T + 41T^{2} \)
43 \( 1 - 1.65T + 43T^{2} \)
47 \( 1 - 11.6T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 7.65T + 59T^{2} \)
61 \( 1 - 13.3T + 61T^{2} \)
67 \( 1 + 6.82T + 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + 0.343T + 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 + 3.65T + 83T^{2} \)
89 \( 1 - 14.8T + 89T^{2} \)
97 \( 1 + 3.65T + 97T^{2} \)
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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

−10.01293362686919311478139673849, −9.126299750289046480372745411639, −8.623147087864520661662925095689, −7.44462319695299937490769520532, −6.56594957851018313229569327837, −5.73721694348781948525770224204, −4.67284196325910695812014020926, −3.93254743488139915975849439666, −2.51711774870063545633816539512, −0.60191173396043412272681005785, 0.60191173396043412272681005785, 2.51711774870063545633816539512, 3.93254743488139915975849439666, 4.67284196325910695812014020926, 5.73721694348781948525770224204, 6.56594957851018313229569327837, 7.44462319695299937490769520532, 8.623147087864520661662925095689, 9.126299750289046480372745411639, 10.01293362686919311478139673849

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