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Properties

Label 2-975-65.29-c1-0-2
Degree $2$
Conductor $975$
Sign $-0.984 - 0.175i$
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  + (−2.21 + 1.28i)2-s + (0.866 − 0.5i)3-s + (2.28 − 3.95i)4-s + (−1.28 + 2.21i)6-s + (−3.08 − 1.78i)7-s + 6.56i·8-s + (0.499 − 0.866i)9-s + (1 + 1.73i)11-s − 4.56i·12-s + (−3.57 + 0.5i)13-s + 9.12·14-s + (−3.84 − 6.65i)16-s + (2.21 + 1.28i)17-s + 2.56i·18-s + (−0.561 + 0.972i)19-s + ⋯
L(s)  = 1  + (−1.56 + 0.905i)2-s + (0.499 − 0.288i)3-s + (1.14 − 1.97i)4-s + (−0.522 + 0.905i)6-s + (−1.16 − 0.673i)7-s + 2.31i·8-s + (0.166 − 0.288i)9-s + (0.301 + 0.522i)11-s − 1.31i·12-s + (−0.990 + 0.138i)13-s + 2.43·14-s + (−0.960 − 1.66i)16-s + (0.538 + 0.310i)17-s + 0.603i·18-s + (−0.128 + 0.223i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.984 - 0.175i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (874, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.984 - 0.175i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0173260 + 0.196137i\)
\(L(\frac12)\) \(\approx\) \(0.0173260 + 0.196137i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.57 - 0.5i)T \)
good2 \( 1 + (2.21 - 1.28i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (3.08 + 1.78i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.21 - 1.28i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.561 - 0.972i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.73 + i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.84 + 4.92i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.56T + 31T^{2} \)
37 \( 1 + (2.97 - 1.71i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.28 + 2.21i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.379 + 0.219i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.24iT - 47T^{2} \)
53 \( 1 - 11.6iT - 53T^{2} \)
59 \( 1 + (5.56 - 9.63i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.06 - 10.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.379 - 0.219i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (7 - 12.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 1.87iT - 73T^{2} \)
79 \( 1 + 9.56T + 79T^{2} \)
83 \( 1 + 9.12iT - 83T^{2} \)
89 \( 1 + (-6.56 - 11.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.84 + 2.21i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.06603925713753206529915815798, −9.424829315529529284928838536226, −8.842066704265665596456159339245, −7.67908436454739046200853846736, −7.32914705887066273524038939241, −6.56872672683833419318024917252, −5.75906984635488713119105157562, −4.20108844550110871773159141747, −2.74478771609876252828346282572, −1.35556102650083274800823086120, 0.14529876784110387933027812094, 1.89884936386260317005403567009, 3.00416945154012827708168868224, 3.47356559858887383868831531015, 5.22242856862290343523912677097, 6.60428328474177044632072566101, 7.40996142255367847364637683121, 8.318633052729225399735009908864, 9.113093496976325736108502357121, 9.508844907136187474899407708891

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