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Properties

Label 2-1170-65.4-c1-0-35
Degree $2$
Conductor $1170$
Sign $-0.606 - 0.795i$
Analytic cond. $9.34249$
Root an. cond. $3.05654$
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 + (−0.499 − 0.866i)4-s + (−1.26 − 1.84i)5-s + (−2.17 − 3.76i)7-s − 0.999·8-s + (−2.22 + 0.178i)10-s + (2.04 + 1.17i)11-s + (−3.18 − 1.69i)13-s − 4.34·14-s + (−0.5 + 0.866i)16-s + (−2.60 + 1.50i)17-s + (0.585 − 0.338i)19-s + (−0.960 + 2.01i)20-s + (2.04 − 1.17i)22-s + (5.58 + 3.22i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.567 − 0.823i)5-s + (−0.821 − 1.42i)7-s − 0.353·8-s + (−0.704 + 0.0563i)10-s + (0.615 + 0.355i)11-s + (−0.883 − 0.469i)13-s − 1.16·14-s + (−0.125 + 0.216i)16-s + (−0.631 + 0.364i)17-s + (0.134 − 0.0776i)19-s + (−0.214 + 0.451i)20-s + (0.435 − 0.251i)22-s + (1.16 + 0.672i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.606 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.606 - 0.795i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ -0.606 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6277833128\)
\(L(\frac12)\) \(\approx\) \(0.6277833128\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 + (1.26 + 1.84i)T \)
13 \( 1 + (3.18 + 1.69i)T \)
good7 \( 1 + (2.17 + 3.76i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.04 - 1.17i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.60 - 1.50i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.585 + 0.338i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.58 - 3.22i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.82 - 8.35i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.11iT - 31T^{2} \)
37 \( 1 + (-3.74 + 6.48i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.60 - 1.50i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.91 - 3.41i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.61T + 47T^{2} \)
53 \( 1 + 9.43iT - 53T^{2} \)
59 \( 1 + (-4.56 + 2.63i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.15 - 3.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.91 - 5.04i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.52 - 1.45i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 7.67T + 73T^{2} \)
79 \( 1 + 3.74T + 79T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 + (4.15 + 2.39i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.17 + 14.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.545066743215879253582836051239, −8.590127714364853257072661147881, −7.38427127130390003005095165787, −6.97613456361520624935907905345, −5.63421456858660865260255650621, −4.62891092916121156391852164909, −3.97352182822856393623716026296, −3.14147148773562638709949672389, −1.47223673817556709181586449324, −0.24210824479611131174410567684, 2.47920367500569096094810180775, 3.16048142878067037385001348142, 4.30796620732110268767826390851, 5.32271695704096431057486192550, 6.39181669639872892829185136254, 6.71817727573123320083764717655, 7.69779790494387532478234148829, 8.734502175509939769172901301262, 9.258638488113208432506904929615, 10.15853037485789551194370294626

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