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Properties

Label 2-1170-65.4-c1-0-34
Degree $2$
Conductor $1170$
Sign $-0.923 - 0.384i$
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 + (0.571 − 2.16i)5-s + (−0.603 − 1.04i)7-s − 0.999·8-s + (−1.58 − 1.57i)10-s + (−4.46 − 2.57i)11-s + (−2.24 + 2.82i)13-s − 1.20·14-s + (−0.5 + 0.866i)16-s + (4.10 − 2.36i)17-s + (−1.84 + 1.06i)19-s + (−2.15 + 0.585i)20-s + (−4.46 + 2.57i)22-s + (−1.88 − 1.08i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.255 − 0.966i)5-s + (−0.227 − 0.394i)7-s − 0.353·8-s + (−0.501 − 0.498i)10-s + (−1.34 − 0.776i)11-s + (−0.622 + 0.782i)13-s − 0.322·14-s + (−0.125 + 0.216i)16-s + (0.994 − 0.574i)17-s + (−0.423 + 0.244i)19-s + (−0.482 + 0.130i)20-s + (−0.951 + 0.549i)22-s + (−0.392 − 0.226i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.384i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 - 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.923 - 0.384i$
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: \((2,\ 1170,\ (\ :1/2),\ -0.923 - 0.384i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9517126640\)
\(L(\frac12)\) \(\approx\) \(0.9517126640\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.571 + 2.16i)T \)
13 \( 1 + (2.24 - 2.82i)T \)
good7 \( 1 + (0.603 + 1.04i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.46 + 2.57i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-4.10 + 2.36i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.84 - 1.06i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.88 + 1.08i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.38 - 4.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.91iT - 31T^{2} \)
37 \( 1 + (-2.20 + 3.81i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.10 + 2.36i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.70 - 0.986i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 0.852T + 47T^{2} \)
53 \( 1 + 4.48iT - 53T^{2} \)
59 \( 1 + (1.68 - 0.970i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.53 + 2.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.02 + 12.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.298 + 0.172i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 15.7T + 73T^{2} \)
79 \( 1 + 13.5T + 79T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 + (14.1 + 8.19i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.24 - 7.34i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.489441200742861732705062972294, −8.599851283753869221859410462126, −7.83103830620384298787706962931, −6.74311919745056748683768424157, −5.50606429923624739196824759398, −5.10685298614411983753933214507, −4.02840059989082806793783520273, −2.97182163260556614033146014628, −1.77995620456304884985291206576, −0.34034536052109821017106239506, 2.31209898832438803099292126497, 3.03747215023952106248551050487, 4.25654297975142928252904984566, 5.45320229559360508871059204446, 5.88881736093636906686747919554, 6.97255210593319010756252958710, 7.71062702467060084050789822402, 8.221849068997413989332270925736, 9.725300805886516188907131227509, 10.00092890309901541051988569336

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