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Properties

Label 2-370-37.7-c1-0-2
Degree $2$
Conductor $370$
Sign $0.885 - 0.465i$
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 + (−2.36 − 0.860i)3-s + (0.766 − 0.642i)4-s + (0.173 + 0.984i)5-s − 2.51·6-s + (0.832 + 4.72i)7-s + (0.500 − 0.866i)8-s + (2.54 + 2.13i)9-s + (0.5 + 0.866i)10-s + (−0.474 + 0.821i)11-s + (−2.36 + 0.860i)12-s + (1.77 − 1.48i)13-s + (2.39 + 4.15i)14-s + (0.436 − 2.47i)15-s + (0.173 − 0.984i)16-s + (4.15 + 3.48i)17-s + ⋯
L(s)  = 1  + (0.664 − 0.241i)2-s + (−1.36 − 0.496i)3-s + (0.383 − 0.321i)4-s + (0.0776 + 0.440i)5-s − 1.02·6-s + (0.314 + 1.78i)7-s + (0.176 − 0.306i)8-s + (0.849 + 0.712i)9-s + (0.158 + 0.273i)10-s + (−0.142 + 0.247i)11-s + (−0.682 + 0.248i)12-s + (0.491 − 0.412i)13-s + (0.640 + 1.10i)14-s + (0.112 − 0.639i)15-s + (0.0434 − 0.246i)16-s + (1.00 + 0.845i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 - 0.465i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.885 - 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.885 - 0.465i$
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: \((2,\ 370,\ (\ :1/2),\ 0.885 - 0.465i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27035 + 0.313407i\)
\(L(\frac12)\) \(\approx\) \(1.27035 + 0.313407i\)
\(L(\frac{3}{2})\) 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 \( 1 + (6.00 + 0.948i)T \)
good3 \( 1 + (2.36 + 0.860i)T + (2.29 + 1.92i)T^{2} \)
7 \( 1 + (-0.832 - 4.72i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (0.474 - 0.821i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.77 + 1.48i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-4.15 - 3.48i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (0.227 + 0.0828i)T + (14.5 + 12.2i)T^{2} \)
23 \( 1 + (-2.20 - 3.81i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.45 + 2.51i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.76T + 31T^{2} \)
41 \( 1 + (-4.90 + 4.11i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + 11.4T + 43T^{2} \)
47 \( 1 + (-3.87 - 6.71i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.821 - 4.65i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (0.848 - 4.81i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-3.83 + 3.21i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (0.406 + 2.30i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (15.7 + 5.71i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 - 8.22T + 73T^{2} \)
79 \( 1 + (1.52 + 8.65i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-5.80 - 4.87i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (-1.15 + 6.53i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (1.02 + 1.78i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.65174630501864429697689131611, −10.93384864727892943879047681967, −10.00344543599315453738265621473, −8.650606654327160457030192030721, −7.43035599880263898734644718580, −6.12194020965920418448740316955, −5.80361929281491458772536116407, −4.93047494532288909337760532397, −3.14441035192117249199668222006, −1.69936822677254912889260579890, 0.935145329617913061237942082265, 3.60010475692540937062953620910, 4.63268884948232609141773274903, 5.20770157909656880272585648476, 6.42726686159806998540650848909, 7.18800357852795638842228571881, 8.365877212808507987520244027939, 9.956904347558438012968520950531, 10.57865038668192000455374407318, 11.37248776568807299541240159028

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