Properties

Label 2-975-1.1-c1-0-30
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  + 2.48·2-s + 3-s + 4.19·4-s + 2.48·6-s + 1.19·7-s + 5.46·8-s + 9-s − 1.19·11-s + 4.19·12-s − 13-s + 2.97·14-s + 5.21·16-s − 6.17·17-s + 2.48·18-s + 6.97·19-s + 1.19·21-s − 2.97·22-s − 4.17·23-s + 5.46·24-s − 2.48·26-s + 27-s + 5.02·28-s + 6·29-s − 2.97·31-s + 2.05·32-s − 1.19·33-s − 15.3·34-s + ⋯
L(s)  = 1  + 1.76·2-s + 0.577·3-s + 2.09·4-s + 1.01·6-s + 0.452·7-s + 1.93·8-s + 0.333·9-s − 0.360·11-s + 1.21·12-s − 0.277·13-s + 0.796·14-s + 1.30·16-s − 1.49·17-s + 0.586·18-s + 1.60·19-s + 0.261·21-s − 0.635·22-s − 0.870·23-s + 1.11·24-s − 0.488·26-s + 0.192·27-s + 0.948·28-s + 1.11·29-s − 0.534·31-s + 0.362·32-s − 0.208·33-s − 2.63·34-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

\(L(1)\) \(\approx\) \(5.322142323\)
\(L(\frac12)\) \(\approx\) \(5.322142323\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.48T + 2T^{2} \)
7 \( 1 - 1.19T + 7T^{2} \)
11 \( 1 + 1.19T + 11T^{2} \)
17 \( 1 + 6.17T + 17T^{2} \)
19 \( 1 - 6.97T + 19T^{2} \)
23 \( 1 + 4.17T + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 2.97T + 31T^{2} \)
37 \( 1 + 7.78T + 37T^{2} \)
41 \( 1 + 6.17T + 41T^{2} \)
43 \( 1 - 9.95T + 43T^{2} \)
47 \( 1 - 1.02T + 47T^{2} \)
53 \( 1 + 10.1T + 53T^{2} \)
59 \( 1 - 5.37T + 59T^{2} \)
61 \( 1 - 12.5T + 61T^{2} \)
67 \( 1 + 9.37T + 67T^{2} \)
71 \( 1 + 5.19T + 71T^{2} \)
73 \( 1 - 11.9T + 73T^{2} \)
79 \( 1 + 1.78T + 79T^{2} \)
83 \( 1 - 5.37T + 83T^{2} \)
89 \( 1 - 10.1T + 89T^{2} \)
97 \( 1 - 1.82T + 97T^{2} \)
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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

−10.24331719941047994473702351253, −9.155284213213173735058270258673, −8.097448163349804344522488895387, −7.23346096847671458642113377924, −6.46994556445513384007715817161, −5.35950812860124869186381001083, −4.72031648974748641580814814830, −3.79404807817002040399353500790, −2.84053070422656402374494518071, −1.90145730813216072519372550098, 1.90145730813216072519372550098, 2.84053070422656402374494518071, 3.79404807817002040399353500790, 4.72031648974748641580814814830, 5.35950812860124869186381001083, 6.46994556445513384007715817161, 7.23346096847671458642113377924, 8.097448163349804344522488895387, 9.155284213213173735058270258673, 10.24331719941047994473702351253

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