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Properties

Label 2-74-37.27-c1-0-3
Degree $2$
Conductor $74$
Sign $-0.783 + 0.621i$
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 + (−1.36 − 2.36i)3-s + (0.499 + 0.866i)4-s + (−1.5 + 0.866i)5-s + 2.73i·6-s + (−2 − 3.46i)7-s − 0.999i·8-s + (−2.23 + 3.86i)9-s + 1.73·10-s + 1.26·11-s + (1.36 − 2.36i)12-s + (5.19 − 3i)13-s + 3.99i·14-s + (4.09 + 2.36i)15-s + (−0.5 + 0.866i)16-s + (−1.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.788 − 1.36i)3-s + (0.249 + 0.433i)4-s + (−0.670 + 0.387i)5-s + 1.11i·6-s + (−0.755 − 1.30i)7-s − 0.353i·8-s + (−0.744 + 1.28i)9-s + 0.547·10-s + 0.382·11-s + (0.394 − 0.683i)12-s + (1.44 − 0.832i)13-s + 1.06i·14-s + (1.05 + 0.610i)15-s + (−0.125 + 0.216i)16-s + (−0.363 − 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.783 + 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.783 + 0.621i$
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: \((2,\ 74,\ (\ :1/2),\ -0.783 + 0.621i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.149926 - 0.430186i\)
\(L(\frac12)\) \(\approx\) \(0.149926 - 0.430186i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 6.06i)T \)
good3 \( 1 + (1.36 + 2.36i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (2 + 3.46i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 1.26T + 11T^{2} \)
13 \( 1 + (-5.19 + 3i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.5 + 0.866i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.09 + 2.36i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.73iT - 23T^{2} \)
29 \( 1 + 7.73iT - 29T^{2} \)
31 \( 1 + 4.73iT - 31T^{2} \)
41 \( 1 + (-3.23 - 5.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 2.53iT - 43T^{2} \)
47 \( 1 + 5.66T + 47T^{2} \)
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 \( 1 + (-2.30 + 1.33i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.09 - 3.63i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.73 + 8.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 8.39T + 73T^{2} \)
79 \( 1 + (1.90 - 1.09i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.83 + 4.90i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.5 - 2.59i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.19iT - 97T^{2} \)
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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

−13.37524522541717917553724457816, −13.28056955072063000266675369900, −11.59107093898854370521530408538, −11.23766587534515271438891961803, −9.853175036921629171389176806789, −7.967186685140303605829202523961, −7.19029689985579873610718444027, −6.20445608735266425771530006515, −3.55463486096395202303844043097, −0.858817157541367596862807090764, 3.84129585771287123298936841095, 5.41074817060273860177902105043, 6.47761877574681070624724372865, 8.675113731009073212394740525707, 9.202167414591515130382824437452, 10.50272125885903166084515845387, 11.52178698610010625578756354788, 12.39673481354989473711917451141, 14.39837071122346000293267736010, 15.67399826770708661959899643735

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