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Properties

Label 2-1710-95.49-c1-0-42
Degree $2$
Conductor $1710$
Sign $0.938 + 0.344i$
Analytic cond. $13.6544$
Root an. cond. $3.69518$
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (0.837 − 2.07i)5-s − 0.785i·7-s + 0.999i·8-s + (1.76 − 1.37i)10-s + 0.377·11-s + (2.51 − 1.45i)13-s + (0.392 − 0.680i)14-s + (−0.5 + 0.866i)16-s + (2.45 + 1.41i)17-s + (−2.82 + 3.32i)19-s + (2.21 − 0.311i)20-s + (0.327 + 0.188i)22-s + (7.86 − 4.54i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.374 − 0.927i)5-s − 0.296i·7-s + 0.353i·8-s + (0.557 − 0.435i)10-s + 0.113·11-s + (0.697 − 0.402i)13-s + (0.104 − 0.181i)14-s + (−0.125 + 0.216i)16-s + (0.595 + 0.343i)17-s + (−0.647 + 0.762i)19-s + (0.495 − 0.0695i)20-s + (0.0697 + 0.0402i)22-s + (1.64 − 0.947i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.344i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 + 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.938 + 0.344i$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1710} (1189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ 0.938 + 0.344i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.760813743\)
\(L(\frac12)\) \(\approx\) \(2.760813743\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
5 \( 1 + (-0.837 + 2.07i)T \)
19 \( 1 + (2.82 - 3.32i)T \)
good7 \( 1 + 0.785iT - 7T^{2} \)
11 \( 1 - 0.377T + 11T^{2} \)
13 \( 1 + (-2.51 + 1.45i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.45 - 1.41i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-7.86 + 4.54i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.01 + 3.48i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.42T + 31T^{2} \)
37 \( 1 - 0.0967iT - 37T^{2} \)
41 \( 1 + (-1.43 + 2.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.371 - 0.214i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-10.4 + 6.04i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.00 + 3.46i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.88 + 8.45i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.27 + 3.94i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.31 - 4.22i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.86 - 10.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.95 - 1.13i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.785 - 1.36i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.75iT - 83T^{2} \)
89 \( 1 + (5.96 + 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.98 + 1.72i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.008590344373619767184605993786, −8.542410170845131967774990843829, −7.68793639664376596443746188452, −6.78419841338222981264141182452, −5.84153193508807336513789694566, −5.34662603429134491567400816814, −4.29318997783420476300793171817, −3.62552662095192748619579946152, −2.26659533989507826961584332687, −0.960762116046706131917259293940, 1.37960991791365526001844095743, 2.58175110487369703960140711710, 3.29764653112622761267629847197, 4.27937176602942670236248501258, 5.41951215081568936224763964382, 5.98376848980677121591046051370, 6.98619986875254373873004378584, 7.42879012810256332516803404881, 8.957387600118801034103962444506, 9.268575625659732802104914879381

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