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Properties

Label 2-975-5.4-c1-0-6
Degree $2$
Conductor $975$
Sign $0.894 + 0.447i$
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  − 2.51i·2-s + i·3-s − 4.32·4-s + 2.51·6-s + 0.514i·7-s + 5.83i·8-s − 9-s − 0.806·11-s − 4.32i·12-s + i·13-s + 1.29·14-s + 6.02·16-s + 2.02i·17-s + 2.51i·18-s − 0.292·19-s + ⋯
L(s)  = 1  − 1.77i·2-s + 0.577i·3-s − 2.16·4-s + 1.02·6-s + 0.194i·7-s + 2.06i·8-s − 0.333·9-s − 0.243·11-s − 1.24i·12-s + 0.277i·13-s + 0.345·14-s + 1.50·16-s + 0.491i·17-s + 0.592i·18-s − 0.0671·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.894 + 0.447i$
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: \((2,\ 975,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08773 - 0.256778i\)
\(L(\frac12)\) \(\approx\) \(1.08773 - 0.256778i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.51iT - 2T^{2} \)
7 \( 1 - 0.514iT - 7T^{2} \)
11 \( 1 + 0.806T + 11T^{2} \)
17 \( 1 - 2.02iT - 17T^{2} \)
19 \( 1 + 0.292T + 19T^{2} \)
23 \( 1 - 8.34iT - 23T^{2} \)
29 \( 1 - 9.64T + 29T^{2} \)
31 \( 1 - 6.22T + 31T^{2} \)
37 \( 1 - 6.34iT - 37T^{2} \)
41 \( 1 + 1.70T + 41T^{2} \)
43 \( 1 + 7.96iT - 43T^{2} \)
47 \( 1 - 10.7iT - 47T^{2} \)
53 \( 1 - 4.70iT - 53T^{2} \)
59 \( 1 + 4.12T + 59T^{2} \)
61 \( 1 + 2.61T + 61T^{2} \)
67 \( 1 + 13.5iT - 67T^{2} \)
71 \( 1 + 7.70T + 71T^{2} \)
73 \( 1 - 11.3iT - 73T^{2} \)
79 \( 1 - 4.73T + 79T^{2} \)
83 \( 1 + 6.57iT - 83T^{2} \)
89 \( 1 - 0.971T + 89T^{2} \)
97 \( 1 + 9.61iT - 97T^{2} \)
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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

−10.22982263170416187211229940440, −9.385823902607537679907626558753, −8.748886316196664134674188401590, −7.82309218603177240424716521398, −6.29373883206021579974589651547, −5.11666255506635906631378459654, −4.36534472871458053532790929914, −3.40885011459525343733159436749, −2.57582534394679469495839714636, −1.28094694012093132392281855061, 0.58978844027376532899880907408, 2.71339314448075267974099156444, 4.31050020533573778479441065923, 5.06588219914287454699865967953, 6.08492744398804243143017131218, 6.70181515307991040136982248303, 7.39670043872703580577406028915, 8.312707058283460082601713502136, 8.656648127395304970105699603882, 9.825823063060250084229888253267

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