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Properties

Label 2-370-37.21-c1-0-7
Degree $2$
Conductor $370$
Sign $0.850 - 0.526i$
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  + (−0.342 + 0.939i)2-s + (3.24 − 1.18i)3-s + (−0.766 − 0.642i)4-s + (0.984 + 0.173i)5-s + 3.45i·6-s + (−0.802 + 4.55i)7-s + (0.866 − 0.500i)8-s + (6.83 − 5.73i)9-s + (−0.5 + 0.866i)10-s + (−0.349 − 0.606i)11-s + (−3.24 − 1.18i)12-s + (−2.84 + 3.39i)13-s + (−4.00 − 2.31i)14-s + (3.40 − 0.599i)15-s + (0.173 + 0.984i)16-s + (−1.72 − 2.06i)17-s + ⋯
L(s)  = 1  + (−0.241 + 0.664i)2-s + (1.87 − 0.681i)3-s + (−0.383 − 0.321i)4-s + (0.440 + 0.0776i)5-s + 1.40i·6-s + (−0.303 + 1.72i)7-s + (0.306 − 0.176i)8-s + (2.27 − 1.91i)9-s + (−0.158 + 0.273i)10-s + (−0.105 − 0.182i)11-s + (−0.936 − 0.340i)12-s + (−0.789 + 0.940i)13-s + (−1.06 − 0.617i)14-s + (0.877 − 0.154i)15-s + (0.0434 + 0.246i)16-s + (−0.419 − 0.499i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.526i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.850 - 0.526i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.850 - 0.526i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.03131 + 0.577616i\)
\(L(\frac12)\) \(\approx\) \(2.03131 + 0.577616i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.342 - 0.939i)T \)
5 \( 1 + (-0.984 - 0.173i)T \)
37 \( 1 + (5.33 + 2.92i)T \)
good3 \( 1 + (-3.24 + 1.18i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (0.802 - 4.55i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (0.349 + 0.606i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.84 - 3.39i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (1.72 + 2.06i)T + (-2.95 + 16.7i)T^{2} \)
19 \( 1 + (0.965 + 2.65i)T + (-14.5 + 12.2i)T^{2} \)
23 \( 1 + (3.45 + 1.99i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.19 + 1.84i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.25iT - 31T^{2} \)
41 \( 1 + (-6.98 - 5.86i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + 2.50iT - 43T^{2} \)
47 \( 1 + (4.65 - 8.05i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.0217 - 0.123i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (-0.819 + 0.144i)T + (55.4 - 20.1i)T^{2} \)
61 \( 1 + (1.85 - 2.20i)T + (-10.5 - 60.0i)T^{2} \)
67 \( 1 + (1.45 - 8.27i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-3.08 + 1.12i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + 4.53T + 73T^{2} \)
79 \( 1 + (9.31 + 1.64i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (1.52 - 1.28i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (-0.886 + 0.156i)T + (83.6 - 30.4i)T^{2} \)
97 \( 1 + (-13.1 - 7.61i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.79451595070348736714489423214, −9.857831175772308066729309808234, −9.249963180140357054905923239543, −8.780444737989553009981229924985, −7.891807788396148645143166610405, −6.87350396915702463082999835207, −6.07385658132718518287367984621, −4.46519084596470775765474102312, −2.75476159765869590829832291840, −2.11299691117919043779833386591, 1.74747218039184994068121122479, 3.09015515176339869177734492422, 3.88569539011535471115523178879, 4.82748733347837818512588744051, 7.07680350642931821846229126914, 7.85260420721790030485476448174, 8.651129812271473952810117081977, 9.732619948000887391121720736911, 10.27501852600806947582607429811, 10.60500375340435595713653075962

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