Properties

Label 2-1170-65.4-c1-0-1
Degree $2$
Conductor $1170$
Sign $-0.408 - 0.912i$
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 + (−2.03 − 0.928i)5-s + (1.40 + 2.42i)7-s − 0.999·8-s + (−1.82 + 1.29i)10-s + (0.515 + 0.297i)11-s + (−1.10 − 3.43i)13-s + 2.80·14-s + (−0.5 + 0.866i)16-s + (−4.98 + 2.87i)17-s + (−6.59 + 3.80i)19-s + (0.213 + 2.22i)20-s + (0.515 − 0.297i)22-s + (−4.02 − 2.32i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.909 − 0.415i)5-s + (0.530 + 0.918i)7-s − 0.353·8-s + (−0.575 + 0.410i)10-s + (0.155 + 0.0896i)11-s + (−0.307 − 0.951i)13-s + 0.749·14-s + (−0.125 + 0.216i)16-s + (−1.20 + 0.697i)17-s + (−1.51 + 0.873i)19-s + (0.0476 + 0.497i)20-s + (0.109 − 0.0634i)22-s + (−0.838 − 0.484i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.408 - 0.912i$
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: \((2,\ 1170,\ (\ :1/2),\ -0.408 - 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2239967775\)
\(L(\frac12)\) \(\approx\) \(0.2239967775\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.03 + 0.928i)T \)
13 \( 1 + (1.10 + 3.43i)T \)
good7 \( 1 + (-1.40 - 2.42i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.515 - 0.297i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (4.98 - 2.87i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.59 - 3.80i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.02 + 2.32i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.26 + 2.18i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.59iT - 31T^{2} \)
37 \( 1 + (5.18 - 8.98i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.98 - 2.87i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.67 - 2.12i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.89T + 47T^{2} \)
53 \( 1 - 13.8iT - 53T^{2} \)
59 \( 1 + (8.40 - 4.85i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.41 + 5.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.93 - 6.80i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.11 + 0.642i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 14.5T + 73T^{2} \)
79 \( 1 + 1.83T + 79T^{2} \)
83 \( 1 + 4.19T + 83T^{2} \)
89 \( 1 + (-5.24 - 3.02i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.45 + 14.6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.24631691675634293937507393929, −9.156675242825747696154945638069, −8.289895865128254175112059295716, −8.016813241295585207325695114692, −6.48439323349400875899136056599, −5.71559139230292687975146702547, −4.58639463162988163781059307267, −4.10753556391377690020442208917, −2.78495414123067486868716045592, −1.74564406225878975558282605968, 0.081576532294861319130142840177, 2.18098037825003723123331940532, 3.64504316890316113365444638135, 4.34992682816306997586199709269, 4.97914745495362151509713723041, 6.60043160900001630510165301779, 6.88141879348985709754525644683, 7.69125481124892941374672875370, 8.574974666579808105326793854636, 9.246965766348929054596029863732

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