↑

Properties

Label 2-975-13.9-c1-0-12
Degree $2$
Conductor $975$
Sign $-0.227 - 0.973i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.813 + 1.40i)2-s + (−0.5 + 0.866i)3-s + (−0.322 − 0.558i)4-s + (−0.813 − 1.40i)6-s + (−1.54 − 2.67i)7-s − 2.20·8-s + (−0.499 − 0.866i)9-s + (1.60 − 2.78i)11-s + 0.644·12-s + (1.88 − 3.07i)13-s + 5.01·14-s + (2.43 − 4.22i)16-s + (2.01 + 3.48i)17-s + 1.62·18-s + (3.66 + 6.34i)19-s + ⋯
L(s)  = 1  + (−0.574 + 0.995i)2-s + (−0.288 + 0.499i)3-s + (−0.161 − 0.279i)4-s + (−0.331 − 0.574i)6-s + (−0.582 − 1.00i)7-s − 0.779·8-s + (−0.166 − 0.288i)9-s + (0.484 − 0.838i)11-s + 0.186·12-s + (0.522 − 0.852i)13-s + 1.34·14-s + (0.609 − 1.05i)16-s + (0.487 + 0.844i)17-s + 0.383·18-s + (0.840 + 1.45i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.227 - 0.973i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.227 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.599950 + 0.755980i\)
\(L(\frac12)\) \(\approx\) \(0.599950 + 0.755980i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 \)
13 \( 1 + (-1.88 + 3.07i)T \)
good2 \( 1 + (0.813 - 1.40i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (1.54 + 2.67i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.60 + 2.78i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.01 - 3.48i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.66 - 6.34i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.77 - 6.54i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.07 - 3.58i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.70T + 31T^{2} \)
37 \( 1 + (2.38 - 4.13i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.01 + 3.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.560 + 0.971i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.97T + 47T^{2} \)
53 \( 1 - 10.0T + 53T^{2} \)
59 \( 1 + (-2.30 - 3.98i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.51 + 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.28 + 7.42i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.60 - 6.24i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 11.9T + 73T^{2} \)
79 \( 1 + 6.12T + 79T^{2} \)
83 \( 1 - 7.24T + 83T^{2} \)
89 \( 1 + (3.04 - 5.27i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.34 + 9.25i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.09174634412610896988791222957, −9.416818917326604801727038806910, −8.343202743784533675298623995249, −7.83531643834330492963951137537, −6.89192345248739009061760223404, −5.95594867245073293918104514336, −5.55470967656795747801387218568, −3.67877709986099843422290326715, −3.48175665404339414113588811360, −0.955117143927324254730013251651, 0.77397962645078541709236786322, 2.18591502973989022224033801254, 2.80593535314417681356980260167, 4.29197538892884274675009956619, 5.57260782418147252302397288458, 6.41538216299602069200083008095, 7.13159232447235800560482403795, 8.451325957506390989511290009735, 9.244552930924283878887337141071, 9.639579424207881931765847356073

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