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Properties

Label 2-370-185.64-c1-0-14
Degree $2$
Conductor $370$
Sign $-0.550 + 0.835i$
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 + (−0.802 + 0.463i)3-s + (−0.499 − 0.866i)4-s + (−0.214 − 2.22i)5-s + 0.926i·6-s + (3.13 − 1.81i)7-s − 0.999·8-s + (−1.07 + 1.85i)9-s + (−2.03 − 0.927i)10-s − 2.25·11-s + (0.802 + 0.463i)12-s + (−2.51 − 4.35i)13-s − 3.62i·14-s + (1.20 + 1.68i)15-s + (−0.5 + 0.866i)16-s + (0.941 − 1.63i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.463 + 0.267i)3-s + (−0.249 − 0.433i)4-s + (−0.0959 − 0.995i)5-s + 0.378i·6-s + (1.18 − 0.684i)7-s − 0.353·8-s + (−0.356 + 0.618i)9-s + (−0.643 − 0.293i)10-s − 0.680·11-s + (0.231 + 0.133i)12-s + (−0.697 − 1.20i)13-s − 0.967i·14-s + (0.310 + 0.435i)15-s + (−0.125 + 0.216i)16-s + (0.228 − 0.395i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.550 + 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.550 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.550 + 0.835i$
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: \((2,\ 370,\ (\ :1/2),\ -0.550 + 0.835i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.589884 - 1.09479i\)
\(L(\frac12)\) \(\approx\) \(0.589884 - 1.09479i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.214 + 2.22i)T \)
37 \( 1 + (-5.63 - 2.28i)T \)
good3 \( 1 + (0.802 - 0.463i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-3.13 + 1.81i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 2.25T + 11T^{2} \)
13 \( 1 + (2.51 + 4.35i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.941 + 1.63i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.77 + 2.75i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 3.62T + 23T^{2} \)
29 \( 1 + 0.243iT - 29T^{2} \)
31 \( 1 + 6.15iT - 31T^{2} \)
41 \( 1 + (-4.85 - 8.40i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 2.54T + 43T^{2} \)
47 \( 1 - 4.91iT - 47T^{2} \)
53 \( 1 + (-10.6 - 6.13i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.09 + 0.632i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.70 + 3.87i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.14 + 2.96i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.93 - 5.08i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 13.0iT - 73T^{2} \)
79 \( 1 + (-9.17 + 5.29i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.77 + 4.49i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-11.0 - 6.36i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 16.1T + 97T^{2} \)
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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

−11.27963780578322769264679671392, −10.31900912517889572222374774626, −9.566882221506278174795397908065, −8.031716676068113204798775439072, −7.75684763903007726557600964236, −5.61073563024316824160703428893, −5.06688456830360166531211616520, −4.32399583153699362388634481452, −2.56933877424371588213501252712, −0.817603354408658210194352680310, 2.22194744295710784921686144164, 3.72331493190949082087332837956, 5.14704955258972727192337764370, 5.89737254135865570292169568423, 6.95199751515157861967401499402, 7.72339401814670235882420660734, 8.722242716295141991983256640444, 9.896078702825352543592691755823, 11.07291344740123655452690223465, 11.88836229834099969083515544552

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