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Properties

Label 2-370-185.99-c1-0-19
Degree $2$
Conductor $370$
Sign $-0.536 + 0.844i$
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  + (0.766 − 0.642i)2-s + (1.56 − 1.86i)3-s + (0.173 − 0.984i)4-s + (−2.11 − 0.731i)5-s − 2.43i·6-s + (−0.424 − 1.16i)7-s + (−0.500 − 0.866i)8-s + (−0.507 − 2.87i)9-s + (−2.08 + 0.797i)10-s + (2.05 + 3.56i)11-s + (−1.56 − 1.86i)12-s + (0.329 − 1.86i)13-s + (−1.07 − 0.619i)14-s + (−4.66 + 2.79i)15-s + (−0.939 − 0.342i)16-s + (−0.180 − 1.02i)17-s + ⋯
L(s)  = 1  + (0.541 − 0.454i)2-s + (0.903 − 1.07i)3-s + (0.0868 − 0.492i)4-s + (−0.944 − 0.327i)5-s − 0.993i·6-s + (−0.160 − 0.440i)7-s + (−0.176 − 0.306i)8-s + (−0.169 − 0.959i)9-s + (−0.660 + 0.252i)10-s + (0.620 + 1.07i)11-s + (−0.451 − 0.538i)12-s + (0.0912 − 0.517i)13-s + (−0.286 − 0.165i)14-s + (−1.20 + 0.721i)15-s + (−0.234 − 0.0855i)16-s + (−0.0437 − 0.248i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.536 + 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.536 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.536 + 0.844i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.536 + 0.844i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.970033 - 1.76563i\)
\(L(\frac12)\) \(\approx\) \(0.970033 - 1.76563i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 + 0.642i)T \)
5 \( 1 + (2.11 + 0.731i)T \)
37 \( 1 + (-5.51 + 2.56i)T \)
good3 \( 1 + (-1.56 + 1.86i)T + (-0.520 - 2.95i)T^{2} \)
7 \( 1 + (0.424 + 1.16i)T + (-5.36 + 4.49i)T^{2} \)
11 \( 1 + (-2.05 - 3.56i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.329 + 1.86i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (0.180 + 1.02i)T + (-15.9 + 5.81i)T^{2} \)
19 \( 1 + (0.677 - 0.807i)T + (-3.29 - 18.7i)T^{2} \)
23 \( 1 + (-2.11 + 3.65i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.476 + 0.274i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.00iT - 31T^{2} \)
41 \( 1 + (1.42 - 8.07i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 - 2.23T + 43T^{2} \)
47 \( 1 + (4.83 + 2.78i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.218 - 0.601i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (1.41 - 3.89i)T + (-45.1 - 37.9i)T^{2} \)
61 \( 1 + (-2.17 - 0.383i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (2.17 + 5.97i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (-11.2 - 9.44i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 - 2.05iT - 73T^{2} \)
79 \( 1 + (5.78 + 15.8i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (17.4 - 3.07i)T + (77.9 - 28.3i)T^{2} \)
89 \( 1 + (2.12 - 5.84i)T + (-68.1 - 57.2i)T^{2} \)
97 \( 1 + (4.25 - 7.36i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.38042788629403604696067871993, −10.25626822016022810607475753872, −9.111045049442387268913200251794, −8.165139615689675561901614125999, −7.30914108457545586755587932994, −6.58752526649666481791443157001, −4.84496846091701062065273497825, −3.82024961671474495670958432824, −2.65504041970898504495137966929, −1.19297369686541547503254854941, 2.87781947845218273984684754851, 3.71170529841721226059506750873, 4.44833830559324578971909464606, 5.85019443289771297307942679569, 6.98701444685639376138261052037, 8.185377187566979512897753752672, 8.798872358981395264197752904484, 9.644784363070580069356778814003, 10.98453269080379124674161312383, 11.56520043970490013185278944433

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