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Properties

Label 2-370-37.7-c1-0-9
Degree $2$
Conductor $370$
Sign $0.999 + 0.00467i$
Analytic cond. $2.95446$
Root an. cond. $1.71885$
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  + (0.939 − 0.342i)2-s + (2.24 + 0.816i)3-s + (0.766 − 0.642i)4-s + (0.173 + 0.984i)5-s + 2.38·6-s + (−0.475 − 2.69i)7-s + (0.500 − 0.866i)8-s + (2.06 + 1.73i)9-s + (0.5 + 0.866i)10-s + (−2.92 + 5.07i)11-s + (2.24 − 0.816i)12-s + (3.89 − 3.26i)13-s + (−1.36 − 2.36i)14-s + (−0.414 + 2.35i)15-s + (0.173 − 0.984i)16-s + (−1.15 − 0.973i)17-s + ⋯
L(s)  = 1  + (0.664 − 0.241i)2-s + (1.29 + 0.471i)3-s + (0.383 − 0.321i)4-s + (0.0776 + 0.440i)5-s + 0.974·6-s + (−0.179 − 1.01i)7-s + (0.176 − 0.306i)8-s + (0.689 + 0.578i)9-s + (0.158 + 0.273i)10-s + (−0.883 + 1.52i)11-s + (0.647 − 0.235i)12-s + (1.08 − 0.906i)13-s + (−0.365 − 0.633i)14-s + (−0.107 + 0.607i)15-s + (0.0434 − 0.246i)16-s + (−0.281 − 0.236i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.999 + 0.00467i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.999 + 0.00467i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.74426 - 0.00641477i\)
\(L(\frac12)\) \(\approx\) \(2.74426 - 0.00641477i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 + 0.342i)T \)
5 \( 1 + (-0.173 - 0.984i)T \)
37 \( 1 + (4.86 + 3.65i)T \)
good3 \( 1 + (-2.24 - 0.816i)T + (2.29 + 1.92i)T^{2} \)
7 \( 1 + (0.475 + 2.69i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (2.92 - 5.07i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.89 + 3.26i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (1.15 + 0.973i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (6.25 + 2.27i)T + (14.5 + 12.2i)T^{2} \)
23 \( 1 + (-3.10 - 5.38i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.30 - 7.45i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.23T + 31T^{2} \)
41 \( 1 + (-1.39 + 1.16i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 - 1.26T + 43T^{2} \)
47 \( 1 + (1.93 + 3.35i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.432 + 2.45i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-1.82 + 10.3i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-4.75 + 3.98i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-1.89 - 10.7i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (9.53 + 3.47i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 - 2.06T + 73T^{2} \)
79 \( 1 + (-2.80 - 15.9i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-0.660 - 0.554i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (0.0179 - 0.102i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (1.62 + 2.81i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.09136220433631424561806955771, −10.48752471748284424058514037179, −9.761305434856690738922659559336, −8.667030238416364003957508842157, −7.54506066117833786048545218871, −6.83840888618760647568473911759, −5.22922809992920857545738163343, −4.05380258550639231710629967736, −3.29095039093761458684029891024, −2.11303501954805228170562290175, 2.07489112092184664386693275464, 3.04012081321797109291831765986, 4.22257206653794253229361249217, 5.76001675263115654488889913107, 6.41339016428214991474467056085, 7.927953667092105846459695040640, 8.655276557452140466969529305270, 8.916341536776422673742610702219, 10.62370537089765402912814083109, 11.58589202196915247007514148202

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