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Properties

Label 2-975-13.3-c1-0-12
Degree $2$
Conductor $975$
Sign $-0.980 - 0.198i$
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  + (1.30 + 2.25i)2-s + (−0.5 − 0.866i)3-s + (−2.38 + 4.12i)4-s + (1.30 − 2.25i)6-s + (1.80 − 3.11i)7-s − 7.20·8-s + (−0.499 + 0.866i)9-s + (2.60 + 4.50i)11-s + 4.76·12-s + (−3.01 + 1.97i)13-s + 9.37·14-s + (−4.60 − 7.97i)16-s + (−1.46 + 2.54i)17-s − 2.60·18-s + (−3.38 + 5.86i)19-s + ⋯
L(s)  = 1  + (0.919 + 1.59i)2-s + (−0.288 − 0.499i)3-s + (−1.19 + 2.06i)4-s + (0.531 − 0.919i)6-s + (0.680 − 1.17i)7-s − 2.54·8-s + (−0.166 + 0.288i)9-s + (0.784 + 1.35i)11-s + 1.37·12-s + (−0.837 + 0.547i)13-s + 2.50·14-s + (−1.15 − 1.99i)16-s + (−0.356 + 0.616i)17-s − 0.613·18-s + (−0.776 + 1.34i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.198i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.980 - 0.198i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.980 - 0.198i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.198666 + 1.97881i\)
\(L(\frac12)\) \(\approx\) \(0.198666 + 1.97881i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.01 - 1.97i)T \)
good2 \( 1 + (-1.30 - 2.25i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-1.80 + 3.11i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.60 - 4.50i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.46 - 2.54i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.38 - 5.86i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.76 - 4.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.916 + 1.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.10T + 31T^{2} \)
37 \( 1 + (1.76 + 3.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.68 - 4.65i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.58 + 2.74i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.80T + 47T^{2} \)
53 \( 1 - 5.20T + 53T^{2} \)
59 \( 1 + (-3.68 + 6.38i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.71 + 2.97i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.75 - 3.03i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.85 + 8.40i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 0.805T + 73T^{2} \)
79 \( 1 + 4.10T + 79T^{2} \)
83 \( 1 - 11.5T + 83T^{2} \)
89 \( 1 + (4.91 + 8.51i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.78 - 4.82i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.32633887725608975146775772159, −9.326174714605778660134519017227, −8.148004007203415320752323046622, −7.57572528245908904956706944377, −6.93233003067832943234293234874, −6.35968255585930713328056824622, −5.21406568882226661792022038801, −4.39168226102659178665561890747, −3.87344771411404240624886983742, −1.79924103358700811487615868683, 0.72409180687371261564408325168, 2.39791753880214058737212629238, 2.98509753553360353882970103703, 4.25633892129612406369379402893, 5.00428889604990079755254481385, 5.61775197030525256456645081557, 6.64302548417492336497714563129, 8.566404610455475445459369046953, 8.971582010329352758076090964582, 9.871306288608725085863255767086

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