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Properties

Label 2-2970-99.32-c1-0-5
Degree $2$
Conductor $2970$
Sign $0.997 + 0.0767i$
Analytic cond. $23.7155$
Root an. cond. $4.86986$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.866 − 0.5i)5-s + (−4.54 + 2.62i)7-s + 0.999·8-s + 0.999i·10-s + (−0.660 − 3.25i)11-s + (−5.39 − 3.11i)13-s + (4.54 + 2.62i)14-s + (−0.5 − 0.866i)16-s − 4.35·17-s − 0.748i·19-s + (0.866 − 0.499i)20-s + (−2.48 + 2.19i)22-s + (−2.87 − 1.66i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.387 − 0.223i)5-s + (−1.71 + 0.992i)7-s + 0.353·8-s + 0.316i·10-s + (−0.199 − 0.979i)11-s + (−1.49 − 0.863i)13-s + (1.21 + 0.701i)14-s + (−0.125 − 0.216i)16-s − 1.05·17-s − 0.171i·19-s + (0.193 − 0.111i)20-s + (−0.529 + 0.468i)22-s + (−0.599 − 0.346i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2970\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 11\)
Sign: $0.997 + 0.0767i$
Analytic conductor: \(23.7155\)
Root analytic conductor: \(4.86986\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2970} (2771, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2970,\ (\ :1/2),\ 0.997 + 0.0767i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3902876969\)
\(L(\frac12)\) \(\approx\) \(0.3902876969\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.660 + 3.25i)T \)
good7 \( 1 + (4.54 - 2.62i)T + (3.5 - 6.06i)T^{2} \)
13 \( 1 + (5.39 + 3.11i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.35T + 17T^{2} \)
19 \( 1 + 0.748iT - 19T^{2} \)
23 \( 1 + (2.87 + 1.66i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.20 - 9.01i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.83 + 4.90i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.09T + 37T^{2} \)
41 \( 1 + (-2.75 + 4.76i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.41 - 0.817i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.640 - 0.369i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.46iT - 53T^{2} \)
59 \( 1 + (-0.877 - 0.506i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.800 - 0.462i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.821 - 1.42i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.56iT - 71T^{2} \)
73 \( 1 + 9.67iT - 73T^{2} \)
79 \( 1 + (10.6 - 6.17i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.86 - 4.95i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 8.41iT - 89T^{2} \)
97 \( 1 + (-8.16 - 14.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.828093701202418327480704179552, −8.286869983389997978216737324718, −7.24220066238063477403452872258, −6.52068761978786277189851543203, −5.64641554030814388770910492193, −4.84937537671092404683592755456, −3.67055282886158528200095003244, −2.90237000031967939115641281383, −2.38850742152672215371411610372, −0.48710291945562862809064214590, 0.28903218323267430316636982612, 2.08610465900797946656888747434, 3.13825158849944033787146139778, 4.30406358962813847507943222554, 4.62720510811688999925020879740, 6.04592218956387603579917010032, 6.78248730965632242597794076172, 7.07778895694224040672927037410, 7.75004953896342945825406114726, 8.779101874752859978152493537575

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