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Properties

Label 2-74-37.36-c1-0-1
Degree $2$
Conductor $74$
Sign $0.753 - 0.657i$
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  + i·2-s + 1.79·3-s − 4-s − 0.791i·5-s + 1.79i·6-s − 2·7-s − i·8-s + 0.208·9-s + 0.791·10-s − 0.791·11-s − 1.79·12-s + 3.79i·13-s − 2i·14-s − 1.41i·15-s + 16-s − 7.58i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.03·3-s − 0.5·4-s − 0.353i·5-s + 0.731i·6-s − 0.755·7-s − 0.353i·8-s + 0.0695·9-s + 0.250·10-s − 0.238·11-s − 0.517·12-s + 1.05i·13-s − 0.534i·14-s − 0.365i·15-s + 0.250·16-s − 1.83i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 - 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.753 - 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.753 - 0.657i$
Analytic conductor: \(0.590892\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :1/2),\ 0.753 - 0.657i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01092 + 0.379142i\)
\(L(\frac12)\) \(\approx\) \(1.01092 + 0.379142i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
37 \( 1 + (-4 - 4.58i)T \)
good3 \( 1 - 1.79T + 3T^{2} \)
5 \( 1 + 0.791iT - 5T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 + 0.791T + 11T^{2} \)
13 \( 1 - 3.79iT - 13T^{2} \)
17 \( 1 + 7.58iT - 17T^{2} \)
19 \( 1 + 1.58iT - 19T^{2} \)
23 \( 1 - 3.79iT - 23T^{2} \)
29 \( 1 + 3.79iT - 29T^{2} \)
31 \( 1 - 8.37iT - 31T^{2} \)
41 \( 1 - 9.79T + 41T^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 + 7.58T + 47T^{2} \)
53 \( 1 + 1.58T + 53T^{2} \)
59 \( 1 + 1.58iT - 59T^{2} \)
61 \( 1 + 12.7iT - 61T^{2} \)
67 \( 1 - 6.37T + 67T^{2} \)
71 \( 1 + 9.16T + 71T^{2} \)
73 \( 1 + 4.37T + 73T^{2} \)
79 \( 1 + 8.20iT - 79T^{2} \)
83 \( 1 - 15.1T + 83T^{2} \)
89 \( 1 + 6iT - 89T^{2} \)
97 \( 1 - 13.5iT - 97T^{2} \)
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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

−14.57429349606826310858075089623, −13.83156345533807426231772609100, −12.98493668412561973418339308322, −11.53096802022679387544575129822, −9.566375257743002105551904355025, −9.079728672022026253730412723331, −7.78092525267977142108962169786, −6.59542030368139108942676692192, −4.84351231959469134717944299118, −3.04893253332420279706449546079, 2.62509516017638999425379875924, 3.77761671350164625409554474866, 5.97832402100446085088432750213, 7.84117264190993255120497014889, 8.822974507471760822583719598981, 10.04927674140541890680663199485, 10.91633864637183167844453857725, 12.64020750102322277673949082459, 13.14215444081536577234990046141, 14.51857667129917132603272307751

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