Properties

Label 2-975-1.1-c1-0-10
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  − 2.77·2-s + 3-s + 5.71·4-s − 2.77·6-s + 2.71·7-s − 10.3·8-s + 9-s − 2.71·11-s + 5.71·12-s − 13-s − 7.55·14-s + 17.2·16-s + 2.83·17-s − 2.77·18-s − 3.55·19-s + 2.71·21-s + 7.55·22-s + 4.83·23-s − 10.3·24-s + 2.77·26-s + 27-s + 15.5·28-s + 6·29-s + 7.55·31-s − 27.3·32-s − 2.71·33-s − 7.88·34-s + ⋯
L(s)  = 1  − 1.96·2-s + 0.577·3-s + 2.85·4-s − 1.13·6-s + 1.02·7-s − 3.65·8-s + 0.333·9-s − 0.820·11-s + 1.65·12-s − 0.277·13-s − 2.01·14-s + 4.31·16-s + 0.688·17-s − 0.654·18-s − 0.816·19-s + 0.593·21-s + 1.61·22-s + 1.00·23-s − 2.10·24-s + 0.544·26-s + 0.192·27-s + 2.93·28-s + 1.11·29-s + 1.35·31-s − 4.83·32-s − 0.473·33-s − 1.35·34-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

\(L(1)\) \(\approx\) \(0.9036100685\)
\(L(\frac12)\) \(\approx\) \(0.9036100685\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.77T + 2T^{2} \)
7 \( 1 - 2.71T + 7T^{2} \)
11 \( 1 + 2.71T + 11T^{2} \)
17 \( 1 - 2.83T + 17T^{2} \)
19 \( 1 + 3.55T + 19T^{2} \)
23 \( 1 - 4.83T + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 7.55T + 31T^{2} \)
37 \( 1 - 4.27T + 37T^{2} \)
41 \( 1 - 2.83T + 41T^{2} \)
43 \( 1 + 11.1T + 43T^{2} \)
47 \( 1 - 11.5T + 47T^{2} \)
53 \( 1 + 1.16T + 53T^{2} \)
59 \( 1 + 2.11T + 59T^{2} \)
61 \( 1 - 6.60T + 61T^{2} \)
67 \( 1 + 1.88T + 67T^{2} \)
71 \( 1 + 6.71T + 71T^{2} \)
73 \( 1 + 9.11T + 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 + 2.11T + 83T^{2} \)
89 \( 1 - 1.16T + 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−10.06309037427633711404424865070, −8.978178205147990256913285490661, −8.378399001020672648982894450179, −7.84261096176926237973580588508, −7.14070728287996220853862685718, −6.12145077375085764124280819691, −4.82334118053611320062057411602, −3.02158224611128967607886093038, −2.19541014199379146725501136288, −1.00003394295883133642957681304, 1.00003394295883133642957681304, 2.19541014199379146725501136288, 3.02158224611128967607886093038, 4.82334118053611320062057411602, 6.12145077375085764124280819691, 7.14070728287996220853862685718, 7.84261096176926237973580588508, 8.378399001020672648982894450179, 8.978178205147990256913285490661, 10.06309037427633711404424865070

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