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Properties

Label 2-975-975.584-c0-0-0
Degree $2$
Conductor $975$
Sign $-0.982 + 0.187i$
Analytic cond. $0.486588$
Root an. cond. $0.697558$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.16 + 1.59i)2-s + (−0.951 + 0.309i)3-s + (−0.896 − 2.76i)4-s + (−0.891 + 0.453i)5-s + (0.610 − 1.87i)6-s + (3.57 + 1.16i)8-s + (0.809 − 0.587i)9-s + (0.309 − 1.95i)10-s + (0.734 + 0.533i)11-s + (1.70 + 2.34i)12-s + (0.587 + 0.809i)13-s + (0.707 − 0.707i)15-s + (−3.65 + 2.65i)16-s + 1.97i·18-s + (2.05 + 2.05i)20-s + ⋯
L(s)  = 1  + (−1.16 + 1.59i)2-s + (−0.951 + 0.309i)3-s + (−0.896 − 2.76i)4-s + (−0.891 + 0.453i)5-s + (0.610 − 1.87i)6-s + (3.57 + 1.16i)8-s + (0.809 − 0.587i)9-s + (0.309 − 1.95i)10-s + (0.734 + 0.533i)11-s + (1.70 + 2.34i)12-s + (0.587 + 0.809i)13-s + (0.707 − 0.707i)15-s + (−3.65 + 2.65i)16-s + 1.97i·18-s + (2.05 + 2.05i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.982 + 0.187i$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (584, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ -0.982 + 0.187i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2929140322\)
\(L(\frac12)\) \(\approx\) \(0.2929140322\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.951 - 0.309i)T \)
5 \( 1 + (0.891 - 0.453i)T \)
13 \( 1 + (-0.587 - 0.809i)T \)
good2 \( 1 + (1.16 - 1.59i)T + (-0.309 - 0.951i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (-0.734 - 0.533i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.309 + 0.951i)T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (0.253 - 0.183i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 - 1.17iT - T^{2} \)
47 \( 1 + (0.297 - 0.0966i)T + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (1.44 - 1.04i)T + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (-0.809 - 0.587i)T^{2} \)
71 \( 1 + (-0.280 - 0.863i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (1.87 + 0.610i)T + (0.809 + 0.587i)T^{2} \)
89 \( 1 + (-1.14 - 0.831i)T + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (-0.809 + 0.587i)T^{2} \)
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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

−10.40349820991945409848137460477, −9.651901022669627423028248204266, −8.929133110313326176445090983642, −8.015104164355377825502598963494, −7.10539920964314420877811143637, −6.63983736146968636087590481322, −5.92509180674870924249692463322, −4.75901844224556011440112636788, −4.08843001413714006069572981953, −1.32959692551193285030458486579, 0.55687960003100934479902586000, 1.64097405616498299695464749173, 3.27809513130539390884624532482, 4.04042384855178557163451991109, 5.10551816626592289244313918911, 6.64843540857111992506591020264, 7.67213745029038932802583783843, 8.277432043605667436676646487498, 9.038502824557011946222610101987, 9.986191108833048499417844489778

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