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Properties

Label 2-370-185.159-c1-0-7
Degree $2$
Conductor $370$
Sign $-0.164 - 0.986i$
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  + (0.5 + 0.866i)2-s + (2.18 + 1.26i)3-s + (−0.499 + 0.866i)4-s + (−2.18 − 0.469i)5-s + 2.52i·6-s + (3 + 1.73i)7-s − 0.999·8-s + (1.68 + 2.92i)9-s + (−0.686 − 2.12i)10-s + (−2.18 + 1.26i)12-s + (−0.186 + 0.322i)13-s + 3.46i·14-s + (−4.18 − 3.78i)15-s + (−0.5 − 0.866i)16-s + (−0.686 − 1.18i)17-s + (−1.68 + 2.92i)18-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (1.26 + 0.728i)3-s + (−0.249 + 0.433i)4-s + (−0.977 − 0.210i)5-s + 1.03i·6-s + (1.13 + 0.654i)7-s − 0.353·8-s + (0.562 + 0.973i)9-s + (−0.216 − 0.672i)10-s + (−0.631 + 0.364i)12-s + (−0.0516 + 0.0894i)13-s + 0.925i·14-s + (−1.08 − 0.977i)15-s + (−0.125 − 0.216i)16-s + (−0.166 − 0.288i)17-s + (−0.397 + 0.688i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.164 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.164 - 0.986i$
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: \((2,\ 370,\ (\ :1/2),\ -0.164 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38353 + 1.63259i\)
\(L(\frac12)\) \(\approx\) \(1.38353 + 1.63259i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.18 + 0.469i)T \)
37 \( 1 + (0.5 + 6.06i)T \)
good3 \( 1 + (-2.18 - 1.26i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-3 - 1.73i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (0.186 - 0.322i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.686 + 1.18i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.74T + 23T^{2} \)
29 \( 1 - 7.72iT - 29T^{2} \)
31 \( 1 + 11.0iT - 31T^{2} \)
41 \( 1 + (-2.87 + 4.97i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 9.11T + 43T^{2} \)
47 \( 1 + 5.04iT - 47T^{2} \)
53 \( 1 + (0.813 - 0.469i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (10.3 - 5.98i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.05 - 1.18i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.11 - 4.10i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.37 - 2.37i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 + (7.11 + 4.10i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-11.7 + 6.78i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (8.31 - 4.80i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.1T + 97T^{2} \)
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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

−11.71413276789180572848175926002, −10.79765651148230883103553561794, −9.347972553490483161062380517552, −8.736685183297613615854632056389, −8.009103367119111803600184230542, −7.32954067897312541375060266821, −5.58702069854528398313571678052, −4.50404571675166366022076083200, −3.80388801610504462774069061791, −2.45035539777414440443180585732, 1.39385357915926162854906098467, 2.73869530870745871558478585155, 3.84612871393757561114006238571, 4.77201742312365643356534033815, 6.60280263546584199037551352732, 7.82241740440227008771759733591, 8.044198352114892466724845792599, 9.147458056774233694967889109879, 10.46544853196073187601241133040, 11.22881039083125986489037982746

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