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Properties

Label 2-975-39.38-c0-0-0
Degree $2$
Conductor $975$
Sign $1$
Analytic cond. $0.486588$
Root an. cond. $0.697558$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s − 3-s + 1.00·4-s + 1.41·6-s + 9-s − 1.41·11-s − 1.00·12-s + 13-s − 0.999·16-s − 1.41·18-s + 2.00·22-s − 1.41·26-s − 27-s + 1.41·32-s + 1.41·33-s + 1.00·36-s − 39-s + 1.41·41-s − 1.41·44-s + 1.41·47-s + 0.999·48-s + 49-s + 1.00·52-s + 1.41·54-s + 1.41·59-s − 1.00·64-s − 2.00·66-s + ⋯
L(s)  = 1  − 1.41·2-s − 3-s + 1.00·4-s + 1.41·6-s + 9-s − 1.41·11-s − 1.00·12-s + 13-s − 0.999·16-s − 1.41·18-s + 2.00·22-s − 1.41·26-s − 27-s + 1.41·32-s + 1.41·33-s + 1.00·36-s − 39-s + 1.41·41-s − 1.41·44-s + 1.41·47-s + 0.999·48-s + 49-s + 1.00·52-s + 1.41·54-s + 1.41·59-s − 1.00·64-s − 2.00·66-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (701, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3372475465\)
\(L(\frac12)\) \(\approx\) \(0.3372475465\)
\(L(1)\) 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 \( 1 + 1.41T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + 1.41T + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - 1.41T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - 1.41T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - 1.41T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
89 \( 1 + 1.41T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.29867747538210091707201643827, −9.478750967385259590983934406778, −8.566426899290467422120468578789, −7.76658190788913118970883933795, −7.08342718096143900721395244712, −6.05022714138797487492329635053, −5.20196199035704887277539312708, −4.05840125957516016357695446664, −2.32328670342634518042683268136, −0.884678411864334047922621702136, 0.884678411864334047922621702136, 2.32328670342634518042683268136, 4.05840125957516016357695446664, 5.20196199035704887277539312708, 6.05022714138797487492329635053, 7.08342718096143900721395244712, 7.76658190788913118970883933795, 8.566426899290467422120468578789, 9.478750967385259590983934406778, 10.29867747538210091707201643827

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