Properties

Label 2-370-185.159-c1-0-14
Degree $2$
Conductor $370$
Sign $-0.894 + 0.446i$
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 + (0.0880 + 0.0508i)3-s + (−0.499 + 0.866i)4-s + (−1.69 − 1.46i)5-s − 0.101i·6-s + (1.94 + 1.12i)7-s + 0.999·8-s + (−1.49 − 2.58i)9-s + (−0.419 + 2.19i)10-s − 3.35·11-s + (−0.0880 + 0.0508i)12-s + (2.04 − 3.54i)13-s − 2.25i·14-s + (−0.0747 − 0.214i)15-s + (−0.5 − 0.866i)16-s + (−0.819 − 1.41i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.0508 + 0.0293i)3-s + (−0.249 + 0.433i)4-s + (−0.756 − 0.653i)5-s − 0.0414i·6-s + (0.736 + 0.425i)7-s + 0.353·8-s + (−0.498 − 0.863i)9-s + (−0.132 + 0.694i)10-s − 1.01·11-s + (−0.0254 + 0.0146i)12-s + (0.568 − 0.984i)13-s − 0.601i·14-s + (−0.0192 − 0.0554i)15-s + (−0.125 − 0.216i)16-s + (−0.198 − 0.344i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.446i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.894 + 0.446i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.894 + 0.446i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.151639 - 0.643004i\)
\(L(\frac12)\) \(\approx\) \(0.151639 - 0.643004i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 + 1.46i)T \)
37 \( 1 + (-3.87 + 4.69i)T \)
good3 \( 1 + (-0.0880 - 0.0508i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-1.94 - 1.12i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 3.35T + 11T^{2} \)
13 \( 1 + (-2.04 + 3.54i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.819 + 1.41i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.13 + 2.38i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.63T + 23T^{2} \)
29 \( 1 + 7.65iT - 29T^{2} \)
31 \( 1 - 1.11iT - 31T^{2} \)
41 \( 1 + (2.65 - 4.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 3.28T + 43T^{2} \)
47 \( 1 - 3.58iT - 47T^{2} \)
53 \( 1 + (0.873 - 0.504i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.82 + 1.63i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.44 - 2.56i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-12.9 - 7.48i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.50 + 2.61i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 9.63iT - 73T^{2} \)
79 \( 1 + (3.11 + 1.80i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.97 + 4.02i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.15 + 2.40i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.86T + 97T^{2} \)
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

−11.21389353730987394083607306616, −10.16219958301750084756791996694, −9.029080923405987947607637738110, −8.279532676223374970817955853235, −7.76168575261113498170843724678, −6.03498186356067052525040354154, −4.89547940239739148754120889769, −3.77699112772650151080570599084, −2.44157610261718983686203392305, −0.48570147145681414295982971779, 2.11191662048776264760291747434, 3.89799300023056099354579885790, 4.95422961870055290315573720877, 6.21296715585374775601380423930, 7.25753175043730799721373097660, 8.118519515419519710867906359643, 8.527417080740982403625544768735, 10.15297405969180063039394082434, 10.85370019314151606944477972093, 11.41247651897687010143245096039

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