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Properties

Label 2-975-65.9-c1-0-2
Degree $2$
Conductor $975$
Sign $-0.0727 - 0.997i$
Analytic cond. $7.78541$
Root an. cond. $2.79023$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (−0.500 − 0.866i)4-s + (0.499 + 0.866i)6-s + (1.73 − i)7-s + 3i·8-s + (0.499 + 0.866i)9-s + (1 − 1.73i)11-s + i·12-s + (−0.866 + 3.5i)13-s − 1.99·14-s + (0.500 − 0.866i)16-s + (−6.06 + 3.5i)17-s − 0.999i·18-s + (−3 − 5.19i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.499 − 0.288i)3-s + (−0.250 − 0.433i)4-s + (0.204 + 0.353i)6-s + (0.654 − 0.377i)7-s + 1.06i·8-s + (0.166 + 0.288i)9-s + (0.301 − 0.522i)11-s + 0.288i·12-s + (−0.240 + 0.970i)13-s − 0.534·14-s + (0.125 − 0.216i)16-s + (−1.47 + 0.848i)17-s − 0.235i·18-s + (−0.688 − 1.19i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0727 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0727 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.0727 - 0.997i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (724, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.0727 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.109383 + 0.117647i\)
\(L(\frac12)\) \(\approx\) \(0.109383 + 0.117647i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 \)
13 \( 1 + (0.866 - 3.5i)T \)
good2 \( 1 + (0.866 + 0.5i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (-1.73 + i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (6.06 - 3.5i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 + 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.19 + 3i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (0.866 + 0.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.19 - 3i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 - 9iT - 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.73 - i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 11iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 14iT - 83T^{2} \)
89 \( 1 + (7 - 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.73 - i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.36922945041336298375562657842, −9.373797175324634487458569381954, −8.643152817132128496172618024184, −7.980642565432569734714684921504, −6.64507643093177295967515543166, −6.18234102616990394911301502534, −4.76767169821405570355701048215, −4.34886751371994639087168668168, −2.34985677678625514229708911618, −1.38991240527399650658459833074, 0.10036757115200319802414967059, 2.01884020059328445376895218204, 3.61760704821086556426416232742, 4.52237486564657571695483921038, 5.43976872868083027408557944012, 6.53939587162967681415380883043, 7.34393297748548440866239240032, 8.277093054494962344925020335085, 8.769601599228892878818899132825, 9.892240624468954381641086330149

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