Properties

Label 2-74-37.26-c1-0-1
Degree $2$
Conductor $74$
Sign $0.0647 - 0.997i$
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  + (0.5 + 0.866i)2-s + (−1 + 1.73i)3-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s − 1.99·6-s − 0.999·8-s + (−0.499 − 0.866i)9-s + 0.999·10-s + 2·11-s + (−0.999 − 1.73i)12-s + (3 − 5.19i)13-s + (0.999 + 1.73i)15-s + (−0.5 − 0.866i)16-s + (−1.5 − 2.59i)17-s + (0.499 − 0.866i)18-s + (−1 + 1.73i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.577 + 0.999i)3-s + (−0.249 + 0.433i)4-s + (0.223 − 0.387i)5-s − 0.816·6-s − 0.353·8-s + (−0.166 − 0.288i)9-s + 0.316·10-s + 0.603·11-s + (−0.288 − 0.499i)12-s + (0.832 − 1.44i)13-s + (0.258 + 0.447i)15-s + (−0.125 − 0.216i)16-s + (−0.363 − 0.630i)17-s + (0.117 − 0.204i)18-s + (−0.229 + 0.397i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0647 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0647 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.0647 - 0.997i$
Analytic conductor: \(0.590892\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :1/2),\ 0.0647 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.697473 + 0.653674i\)
\(L(\frac12)\) \(\approx\) \(0.697473 + 0.653674i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.5 + 6.06i)T \)
good3 \( 1 + (1 - 1.73i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + (-3 + 5.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 - 10T + 31T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 - 2T + 47T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4 + 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3 + 5.19i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + (3 - 5.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1 + 1.73i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.5 - 11.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 3T + 97T^{2} \)
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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

−15.17909447899250491024396027080, −13.83401844787887754104842699083, −12.82618908656418296047681626106, −11.50907661287921777841709860176, −10.36792747105730767268102313947, −9.234038368076750619626841887352, −7.897310048569755711189734443433, −6.11612731662786134262438330044, −5.14059977250709566552519173371, −3.81416583544449140217727258991, 1.81063352494896351087694676103, 4.08846483113855139398748242624, 6.11451485117059988995423096877, 6.79701668017853675435396047183, 8.659543060140600794180498071726, 10.09457197427903449832368084192, 11.45836282369062742200235662188, 11.95528096197997345111936167598, 13.23250139058814471947182016311, 13.91701948656828477372271291806

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