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Properties

Label 2-370-185.8-c1-0-0
Degree $2$
Conductor $370$
Sign $0.800 - 0.598i$
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  + (−0.866 + 0.5i)2-s + (−0.745 − 2.78i)3-s + (0.499 − 0.866i)4-s + (−2.22 + 0.191i)5-s + (2.03 + 2.03i)6-s + (1.31 + 4.90i)7-s + 0.999i·8-s + (−4.58 + 2.64i)9-s + (1.83 − 1.27i)10-s + 0.446i·11-s + (−2.78 − 0.745i)12-s + (4.01 + 2.31i)13-s + (−3.59 − 3.59i)14-s + (2.19 + 6.05i)15-s + (−0.5 − 0.866i)16-s + (0.309 + 0.535i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.430 − 1.60i)3-s + (0.249 − 0.433i)4-s + (−0.996 + 0.0857i)5-s + (0.831 + 0.831i)6-s + (0.496 + 1.85i)7-s + 0.353i·8-s + (−1.52 + 0.881i)9-s + (0.579 − 0.404i)10-s + 0.134i·11-s + (−0.802 − 0.215i)12-s + (1.11 + 0.642i)13-s + (−0.960 − 0.960i)14-s + (0.566 + 1.56i)15-s + (−0.125 − 0.216i)16-s + (0.0750 + 0.129i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.800 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.800 - 0.598i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.800 - 0.598i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.635412 + 0.211254i\)
\(L(\frac12)\) \(\approx\) \(0.635412 + 0.211254i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (2.22 - 0.191i)T \)
37 \( 1 + (-4.81 - 3.72i)T \)
good3 \( 1 + (0.745 + 2.78i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-1.31 - 4.90i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 - 0.446iT - 11T^{2} \)
13 \( 1 + (-4.01 - 2.31i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.309 - 0.535i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.21 + 4.52i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 - 2.27iT - 23T^{2} \)
29 \( 1 + (-4.05 - 4.05i)T + 29iT^{2} \)
31 \( 1 + (-0.176 + 0.176i)T - 31iT^{2} \)
41 \( 1 + (3.38 + 1.95i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 - 8.15iT - 43T^{2} \)
47 \( 1 + (6.46 - 6.46i)T - 47iT^{2} \)
53 \( 1 + (2.16 - 8.08i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-3.85 - 1.03i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.05 - 7.66i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-13.2 + 3.53i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-7.60 + 13.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.847 - 0.847i)T - 73iT^{2} \)
79 \( 1 + (-2.81 - 10.5i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (-2.67 + 9.97i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (0.257 - 0.960i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + 3.22T + 97T^{2} \)
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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

−11.45342119936001574010104252501, −11.14234128357264242456854916330, −9.174298866139386422259898766612, −8.428822956287401726288041274359, −7.87459292156904651196083925281, −6.72657975904980397758108726436, −6.13750121113310762648708620374, −4.95505690905414370339206829026, −2.71103716115852968883488771066, −1.39998219248724335344966949381, 0.65748801966302223085000349813, 3.68055757176997774295664142442, 3.88687317403958736717118251920, 5.03791151674907088852053291983, 6.63760779224693289263273373405, 7.959676199508478515934705718390, 8.465960303725015122956395942688, 9.919885221513878764659869973228, 10.39725478624847143053783447078, 11.08244490035104060171390929396

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