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Properties

Label 2-370-185.104-c1-0-9
Degree $2$
Conductor $370$
Sign $0.959 - 0.282i$
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  + (−0.939 + 0.342i)2-s + (−0.981 + 2.69i)3-s + (0.766 − 0.642i)4-s + (1.03 − 1.98i)5-s − 2.86i·6-s + (2.07 − 0.365i)7-s + (−0.500 + 0.866i)8-s + (−4.00 − 3.36i)9-s + (−0.291 + 2.21i)10-s + (2.94 − 5.09i)11-s + (0.981 + 2.69i)12-s + (4.66 − 3.91i)13-s + (−1.82 + 1.05i)14-s + (4.33 + 4.72i)15-s + (0.173 − 0.984i)16-s + (−5.11 − 4.29i)17-s + ⋯
L(s)  = 1  + (−0.664 + 0.241i)2-s + (−0.566 + 1.55i)3-s + (0.383 − 0.321i)4-s + (0.461 − 0.887i)5-s − 1.17i·6-s + (0.783 − 0.138i)7-s + (−0.176 + 0.306i)8-s + (−1.33 − 1.12i)9-s + (−0.0922 + 0.701i)10-s + (0.887 − 1.53i)11-s + (0.283 + 0.778i)12-s + (1.29 − 1.08i)13-s + (−0.486 + 0.281i)14-s + (1.11 + 1.22i)15-s + (0.0434 − 0.246i)16-s + (−1.24 − 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.282i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.959 - 0.282i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.959 - 0.282i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.966146 + 0.139557i\)
\(L(\frac12)\) \(\approx\) \(0.966146 + 0.139557i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 - 0.342i)T \)
5 \( 1 + (-1.03 + 1.98i)T \)
37 \( 1 + (-2.51 - 5.53i)T \)
good3 \( 1 + (0.981 - 2.69i)T + (-2.29 - 1.92i)T^{2} \)
7 \( 1 + (-2.07 + 0.365i)T + (6.57 - 2.39i)T^{2} \)
11 \( 1 + (-2.94 + 5.09i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.66 + 3.91i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (5.11 + 4.29i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (0.963 - 2.64i)T + (-14.5 - 12.2i)T^{2} \)
23 \( 1 + (-1.71 - 2.96i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.14 - 1.23i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.15iT - 31T^{2} \)
41 \( 1 + (1.23 - 1.03i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + 1.64T + 43T^{2} \)
47 \( 1 + (-8.65 + 4.99i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (9.41 + 1.65i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (6.20 + 1.09i)T + (55.4 + 20.1i)T^{2} \)
61 \( 1 + (-5.62 - 6.70i)T + (-10.5 + 60.0i)T^{2} \)
67 \( 1 + (3.57 - 0.631i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (-7.82 - 2.84i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + 0.888iT - 73T^{2} \)
79 \( 1 + (-3.75 + 0.662i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (-0.0733 + 0.0874i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (-0.148 - 0.0262i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (-9.05 - 15.6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.13148560372275852879908697567, −10.58188204798905158802206108336, −9.500683666397895687835118996700, −8.779905112396399022246778268923, −8.259014896008097831510092997205, −6.31424994607936094517420003520, −5.56441611727642393258541181326, −4.68579981995216215172764344919, −3.45892776358356826615816462872, −0.994137024470520430239470970378, 1.63164531306433778021713743718, 2.15191128287219154356576506469, 4.32593794395309938480134599769, 6.24206394990834147257789312326, 6.61566521769826615214034121024, 7.39722693753941426690048187454, 8.503322379752407538675465816432, 9.416702285917971448735039954649, 10.94540666147229367966241226415, 11.14731330334673633375725048051

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