Properties

Label 2-370-185.162-c1-0-10
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: $\chi_{370} (347, \cdot )$
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

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

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