↑

Properties

Label 2-370-185.88-c1-0-7
Degree $2$
Conductor $370$
Sign $-0.0372 + 0.999i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−2.85 − 0.765i)3-s + (−0.499 − 0.866i)4-s + (2.19 + 0.402i)5-s + (−2.09 + 2.09i)6-s + (3.43 + 0.920i)7-s − 0.999·8-s + (4.98 + 2.87i)9-s + (1.44 − 1.70i)10-s − 4.83i·11-s + (0.765 + 2.85i)12-s + (2.06 + 3.57i)13-s + (2.51 − 2.51i)14-s + (−5.97 − 2.83i)15-s + (−0.5 + 0.866i)16-s + (−6.03 − 3.48i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−1.65 − 0.442i)3-s + (−0.249 − 0.433i)4-s + (0.983 + 0.179i)5-s + (−0.854 + 0.854i)6-s + (1.29 + 0.348i)7-s − 0.353·8-s + (1.66 + 0.959i)9-s + (0.457 − 0.538i)10-s − 1.45i·11-s + (0.221 + 0.825i)12-s + (0.572 + 0.991i)13-s + (0.672 − 0.672i)14-s + (−1.54 − 0.731i)15-s + (−0.125 + 0.216i)16-s + (−1.46 − 0.845i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0372 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0372 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.0372 + 0.999i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.0372 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.851737 - 0.884098i\)
\(L(\frac12)\) \(\approx\) \(0.851737 - 0.884098i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-2.19 - 0.402i)T \)
37 \( 1 + (-4.74 - 3.80i)T \)
good3 \( 1 + (2.85 + 0.765i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-3.43 - 0.920i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + 4.83iT - 11T^{2} \)
13 \( 1 + (-2.06 - 3.57i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (6.03 + 3.48i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.30 + 4.86i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 - 1.47T + 23T^{2} \)
29 \( 1 + (-2.37 + 2.37i)T - 29iT^{2} \)
31 \( 1 + (2.94 + 2.94i)T + 31iT^{2} \)
41 \( 1 + (-1.84 + 1.06i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 - 0.602T + 43T^{2} \)
47 \( 1 + (2.33 - 2.33i)T - 47iT^{2} \)
53 \( 1 + (2.42 - 0.651i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.34 + 0.628i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.69 + 6.31i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-2.39 + 8.93i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-5.05 - 8.75i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (9.08 - 9.08i)T - 73iT^{2} \)
79 \( 1 + (3.81 - 14.2i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (3.67 - 0.985i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-2.51 - 9.39i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 - 11.5iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.18589426488701757836043772082, −11.01715431369837098034160155462, −9.516357474842202910685636215767, −8.565763403382256736018877474918, −6.87693075016475873849992524878, −6.14478872908888740563656898890, −5.27927878900576948084958110472, −4.55424933053308559258753664479, −2.36248378677886196627037173894, −1.05983160344158329912770147907, 1.59733099266038522506950974960, 4.26135688563950975442168320827, 4.93375181560017528519377023334, 5.71066025578554025005386626934, 6.53961068187087157652906124021, 7.61077033854596458675363793183, 8.871174517167550896350823685889, 10.18899635401974357477595319383, 10.63440413854363960111223556010, 11.58110124704625495509989808572

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