Properties

Label 2-975-5.4-c1-0-13
Degree $2$
Conductor $975$
Sign $0.447 - 0.894i$
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  − 0.289i·2-s + i·3-s + 1.91·4-s + 0.289·6-s + 4.91i·7-s − 1.13i·8-s − 9-s + 4.91·11-s + 1.91i·12-s i·13-s + 1.42·14-s + 3.50·16-s − 4.33i·17-s + 0.289i·18-s − 2.57·19-s + ⋯
L(s)  = 1  − 0.204i·2-s + 0.577i·3-s + 0.958·4-s + 0.118·6-s + 1.85i·7-s − 0.400i·8-s − 0.333·9-s + 1.48·11-s + 0.553i·12-s − 0.277i·13-s + 0.379·14-s + 0.876·16-s − 1.05i·17-s + 0.0681i·18-s − 0.591·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77894 + 1.09944i\)
\(L(\frac12)\) \(\approx\) \(1.77894 + 1.09944i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 \)
13 \( 1 + iT \)
good2 \( 1 + 0.289iT - 2T^{2} \)
7 \( 1 - 4.91iT - 7T^{2} \)
11 \( 1 - 4.91T + 11T^{2} \)
17 \( 1 + 4.33iT - 17T^{2} \)
19 \( 1 + 2.57T + 19T^{2} \)
23 \( 1 - 6.33iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 1.42T + 31T^{2} \)
37 \( 1 - 9.49iT - 37T^{2} \)
41 \( 1 - 4.33T + 41T^{2} \)
43 \( 1 - 1.15iT - 43T^{2} \)
47 \( 1 + 5.42iT - 47T^{2} \)
53 \( 1 - 0.338iT - 53T^{2} \)
59 \( 1 - 11.2T + 59T^{2} \)
61 \( 1 + 10.1T + 61T^{2} \)
67 \( 1 + 7.25iT - 67T^{2} \)
71 \( 1 - 0.916T + 71T^{2} \)
73 \( 1 - 3.15iT - 73T^{2} \)
79 \( 1 - 3.49T + 79T^{2} \)
83 \( 1 + 11.2iT - 83T^{2} \)
89 \( 1 - 0.338T + 89T^{2} \)
97 \( 1 + 12.3iT - 97T^{2} \)
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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

−10.02366877665213065412515847565, −9.332503435059081301630500741281, −8.770141544113495597176885643116, −7.66286784669392185369210752828, −6.54701416554594182821886254658, −5.89716524097191964511446655192, −5.04415825064435219545266126214, −3.64011536411592464163751034647, −2.76368165776181243297415920056, −1.71510952892384484279799066133, 1.02974877031961659798242959458, 2.06496727807127902582103824105, 3.66222883397992151075454980071, 4.29599704310334540860921140291, 5.98708061136611726569487958809, 6.61803425444271003609486542931, 7.18038780891415662616736044899, 7.912368761884266965118953834710, 8.888696811246265815702843818650, 10.04824659870449232576108486949

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