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Properties

Label 2-1170-13.10-c1-0-0
Degree $2$
Conductor $1170$
Sign $-0.988 - 0.151i$
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s − i·5-s + (−2.34 − 1.35i)7-s + 0.999i·8-s + (0.5 + 0.866i)10-s + (2.59 − 1.5i)11-s + (−3.34 + 1.35i)13-s + 2.70·14-s + (−0.5 − 0.866i)16-s + (−5.34 − 3.08i)19-s + (−0.866 − 0.499i)20-s + (−1.5 + 2.59i)22-s + (3.95 + 6.84i)23-s − 25-s + (2.21 − 2.84i)26-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s − 0.447i·5-s + (−0.885 − 0.511i)7-s + 0.353i·8-s + (0.158 + 0.273i)10-s + (0.783 − 0.452i)11-s + (−0.926 + 0.375i)13-s + 0.722·14-s + (−0.125 − 0.216i)16-s + (−1.22 − 0.707i)19-s + (−0.193 − 0.111i)20-s + (−0.319 + 0.553i)22-s + (0.823 + 1.42i)23-s − 0.200·25-s + (0.435 − 0.557i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.151i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 - 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.988 - 0.151i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ -0.988 - 0.151i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.01354635469\)
\(L(\frac12)\) \(\approx\) \(0.01354635469\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 \)
5 \( 1 + iT \)
13 \( 1 + (3.34 - 1.35i)T \)
good7 \( 1 + (2.34 + 1.35i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.59 + 1.5i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.34 + 3.08i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.95 - 6.84i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.24 + 2.15i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.972iT - 31T^{2} \)
37 \( 1 + (1.5 - 0.866i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.19 - 3i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.84 - 8.38i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3iT - 47T^{2} \)
53 \( 1 + 2.70T + 53T^{2} \)
59 \( 1 + (6.65 + 3.84i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.68 - 11.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.68 - 0.972i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.19 - 3i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 - 5.68T + 79T^{2} \)
83 \( 1 - 15.3iT - 83T^{2} \)
89 \( 1 + (-4.05 + 2.34i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.68 + 4.43i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.786396227069721662116041537788, −9.435295214524202686134117685249, −8.651534805039490157597598872733, −7.69144435477838325577751536554, −6.82538543903861502223591534525, −6.29405303890924303628765005602, −5.12647066071810875478294164043, −4.14192421956929760140144943303, −2.96695330076044615909540112228, −1.46706725041099350561393169435, 0.00702491351387901617692532418, 1.93668644389090890293232346119, 2.88840592742183879447663382173, 3.86375329101998567512803386481, 5.07940224549387943937229669099, 6.42821375305281040833507040533, 6.77013201130706850433358569785, 7.83130528812283734677104315905, 8.813758742725001422698357261968, 9.352875768343514848973102775003

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