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Properties

Label 2-370-37.11-c1-0-6
Degree $2$
Conductor $370$
Sign $0.546 + 0.837i$
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  + (0.866 − 0.5i)2-s + (0.264 − 0.458i)3-s + (0.499 − 0.866i)4-s + (0.866 + 0.5i)5-s − 0.529i·6-s + (1.80 − 3.11i)7-s − 0.999i·8-s + (1.35 + 2.35i)9-s + 0.999·10-s − 1.82·11-s + (−0.264 − 0.458i)12-s + (−2.25 − 1.30i)13-s − 3.60i·14-s + (0.458 − 0.264i)15-s + (−0.5 − 0.866i)16-s + (−3.42 + 1.97i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.152 − 0.264i)3-s + (0.249 − 0.433i)4-s + (0.387 + 0.223i)5-s − 0.216i·6-s + (0.680 − 1.17i)7-s − 0.353i·8-s + (0.453 + 0.785i)9-s + 0.316·10-s − 0.549·11-s + (−0.0764 − 0.132i)12-s + (−0.624 − 0.360i)13-s − 0.962i·14-s + (0.118 − 0.0683i)15-s + (−0.125 − 0.216i)16-s + (−0.830 + 0.479i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.546 + 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.546 + 0.837i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.546 + 0.837i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90534 - 1.03162i\)
\(L(\frac12)\) \(\approx\) \(1.90534 - 1.03162i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (4.20 - 4.39i)T \)
good3 \( 1 + (-0.264 + 0.458i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-1.80 + 3.11i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 1.82T + 11T^{2} \)
13 \( 1 + (2.25 + 1.30i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.42 - 1.97i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.48 - 3.74i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.10iT - 23T^{2} \)
29 \( 1 + 7.96iT - 29T^{2} \)
31 \( 1 - 3.44iT - 31T^{2} \)
41 \( 1 + (5.37 - 9.31i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 6.69iT - 43T^{2} \)
47 \( 1 + 0.773T + 47T^{2} \)
53 \( 1 + (-0.605 - 1.04i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (9.93 - 5.73i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.69 + 1.55i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.92 - 5.05i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.05 - 7.02i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 11.9T + 73T^{2} \)
79 \( 1 + (-12.0 - 6.97i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.08 + 3.61i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-15.4 + 8.92i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.2iT - 97T^{2} \)
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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

−11.16845495163881528465130124807, −10.38598799379366263686406248334, −9.910180794238483300353392701978, −8.150274005478485403435264625354, −7.51200190977671030518460948333, −6.45971589355692856841334550803, −5.09336339102623139593109006873, −4.35346511750686696119729352152, −2.82946863644594761967556940029, −1.51672010746872932913538164901, 2.10099003952555815849736327222, 3.41521509537541235491232491017, 4.99756577945101958476152997763, 5.33885880731211719019389695275, 6.75139998905118534123996833222, 7.63688247856466239513233339111, 9.093352261155529607008910261617, 9.262165940084086611781901289814, 10.76462322428909712209125116125, 11.87907937842104586996251781881

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