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Properties

Label 2-975-39.38-c0-0-2
Degree $2$
Conductor $975$
Sign $1$
Analytic cond. $0.486588$
Root an. cond. $0.697558$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 4-s + 9-s − 12-s + 13-s + 16-s + 27-s − 36-s + 39-s − 2·43-s + 48-s + 49-s − 52-s − 2·61-s − 64-s − 2·79-s + 81-s − 2·103-s − 108-s + 117-s + ⋯
L(s)  = 1  + 3-s − 4-s + 9-s − 12-s + 13-s + 16-s + 27-s − 36-s + 39-s − 2·43-s + 48-s + 49-s − 52-s − 2·61-s − 64-s − 2·79-s + 81-s − 2·103-s − 108-s + 117-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: yes
Arithmetic: yes
Character: $\chi_{975} (701, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.201133572\)
\(L(\frac12)\) \(\approx\) \(1.201133572\)
\(L(1)\) 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 \( 1 + T^{2} \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( 1 + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( 1 + T^{2} \)
43 \( ( 1 + T )^{2} \)
47 \( 1 + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( 1 + T^{2} \)
61 \( ( 1 + T )^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( 1 + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 + T )^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−10.00635572959259937800495755297, −9.239202745947718578128660267468, −8.576901198018999493202781481617, −8.018180743462944547191474770834, −6.99213629932800525558066459787, −5.85672168271910981191498623139, −4.72795238258038879441389878126, −3.88939227095558166677289173333, −3.05146306244320626223354702587, −1.48345996023349179680320873098, 1.48345996023349179680320873098, 3.05146306244320626223354702587, 3.88939227095558166677289173333, 4.72795238258038879441389878126, 5.85672168271910981191498623139, 6.99213629932800525558066459787, 8.018180743462944547191474770834, 8.576901198018999493202781481617, 9.239202745947718578128660267468, 10.00635572959259937800495755297

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