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Properties

Label 2-74-37.27-c1-0-0
Degree $2$
Conductor $74$
Sign $0.991 + 0.128i$
Analytic cond. $0.590892$
Root an. cond. $0.768695$
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 + (0.366 + 0.633i)3-s + (0.499 + 0.866i)4-s + (1.5 − 0.866i)5-s − 0.732i·6-s + (1 + 1.73i)7-s − 0.999i·8-s + (1.23 − 2.13i)9-s − 1.73·10-s − 1.26·11-s + (−0.366 + 0.633i)12-s + (−3 + 1.73i)13-s − 1.99i·14-s + (1.09 + 0.633i)15-s + (−0.5 + 0.866i)16-s + (−3.69 − 2.13i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.211 + 0.366i)3-s + (0.249 + 0.433i)4-s + (0.670 − 0.387i)5-s − 0.298i·6-s + (0.377 + 0.654i)7-s − 0.353i·8-s + (0.410 − 0.711i)9-s − 0.547·10-s − 0.382·11-s + (−0.105 + 0.183i)12-s + (−0.832 + 0.480i)13-s − 0.534i·14-s + (0.283 + 0.163i)15-s + (−0.125 + 0.216i)16-s + (−0.896 − 0.517i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.128i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.991 + 0.128i$
Analytic conductor: \(0.590892\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :1/2),\ 0.991 + 0.128i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.820702 - 0.0530317i\)
\(L(\frac12)\) \(\approx\) \(0.820702 - 0.0530317i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
37 \( 1 + (-4.69 + 3.86i)T \)
good3 \( 1 + (-0.366 - 0.633i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 1.26T + 11T^{2} \)
13 \( 1 + (3 - 1.73i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.69 + 2.13i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.09 - 2.36i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.26iT - 23T^{2} \)
29 \( 1 + 8.66iT - 29T^{2} \)
31 \( 1 + 4.73iT - 31T^{2} \)
41 \( 1 + (-1.96 - 3.40i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 12.9iT - 43T^{2} \)
47 \( 1 - 1.26T + 47T^{2} \)
53 \( 1 + (4.73 - 8.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (8.19 + 4.73i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.5 + 0.866i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.0980 + 0.169i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.73 - 3i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 + (-14.4 + 8.36i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.83 - 10.0i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-14.8 - 8.59i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.26iT - 97T^{2} \)
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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

−14.75858495202708786681372002521, −13.30788908070663736100767956028, −12.28868372161382340126213867992, −11.18088842630252293402090902745, −9.687449684183083915423721605071, −9.273668859792150488751516843853, −7.892931623663597501548689123524, −6.20786681146086478404619554734, −4.46680475983487499123560289942, −2.27892757616353307373626938367, 2.19017534756175165655994352901, 4.89944297394910492360154880536, 6.61877935023988811205840373381, 7.57989835563077901132931397111, 8.757677978369228472229554727719, 10.34910272065349020369379331293, 10.72979023045933942202522845319, 12.64872893338517397460653238074, 13.66638076376307963233847255616, 14.57120847578038443748412937748

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