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Properties

Label 2-370-185.64-c1-0-15
Degree $2$
Conductor $370$
Sign $0.650 + 0.759i$
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 + (2.50 − 1.44i)3-s + (−0.499 − 0.866i)4-s + (−0.791 − 2.09i)5-s + 2.88i·6-s + (0.668 − 0.385i)7-s + 0.999·8-s + (2.67 − 4.63i)9-s + (2.20 + 0.359i)10-s − 4.03·11-s + (−2.50 − 1.44i)12-s + (1.92 + 3.32i)13-s + 0.771i·14-s + (−5.00 − 4.08i)15-s + (−0.5 + 0.866i)16-s + (3.69 − 6.39i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (1.44 − 0.834i)3-s + (−0.249 − 0.433i)4-s + (−0.354 − 0.935i)5-s + 1.17i·6-s + (0.252 − 0.145i)7-s + 0.353·8-s + (0.891 − 1.54i)9-s + (0.697 + 0.113i)10-s − 1.21·11-s + (−0.722 − 0.417i)12-s + (0.532 + 0.922i)13-s + 0.206i·14-s + (−1.29 − 1.05i)15-s + (−0.125 + 0.216i)16-s + (0.895 − 1.55i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.650 + 0.759i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.650 + 0.759i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47971 - 0.680779i\)
\(L(\frac12)\) \(\approx\) \(1.47971 - 0.680779i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.791 + 2.09i)T \)
37 \( 1 + (4.87 - 3.63i)T \)
good3 \( 1 + (-2.50 + 1.44i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-0.668 + 0.385i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 4.03T + 11T^{2} \)
13 \( 1 + (-1.92 - 3.32i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.69 + 6.39i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.31 - 1.33i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.28T + 23T^{2} \)
29 \( 1 + 3.03iT - 29T^{2} \)
31 \( 1 - 0.197iT - 31T^{2} \)
41 \( 1 + (-2.56 - 4.44i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 7.21T + 43T^{2} \)
47 \( 1 - 1.03iT - 47T^{2} \)
53 \( 1 + (-9.81 - 5.66i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.08 - 3.51i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.52 + 3.19i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.06 - 2.92i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.15 - 12.3i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 3.22iT - 73T^{2} \)
79 \( 1 + (6.11 - 3.52i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (12.9 + 7.48i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.24 + 3.60i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 15.2T + 97T^{2} \)
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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

−11.36356274052847237585995096158, −9.889273726594668408702470507227, −9.035591274396462831516679129213, −8.389168896443556246088935478175, −7.67002218914576968572841291690, −7.02211762385594166445207461230, −5.46725768026707435101311380241, −4.25623710336393391743703089819, −2.72109189420862597743514295857, −1.19466840850384679552288940459, 2.29200397432959423878762496856, 3.22150619061519668809305359347, 3.93974157763486471590686403694, 5.45401591593596451409869070262, 7.29042504773790278383430729295, 8.211923880791137262314076511901, 8.591418798785569287663418042706, 9.902654519800906091710946373996, 10.56820960892938701755353950645, 10.93792827698852063016525313009

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