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Properties

Label 2-1183-7.2-c1-0-45
Degree $2$
Conductor $1183$
Sign $-0.0513 - 0.998i$
Analytic cond. $9.44630$
Root an. cond. $3.07348$
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  + (0.632 − 1.09i)2-s + (1.31 + 2.27i)3-s + (0.199 + 0.344i)4-s + (−1.45 + 2.51i)5-s + 3.32·6-s + (1.29 + 2.30i)7-s + 3.03·8-s + (−1.95 + 3.37i)9-s + (1.83 + 3.18i)10-s + (1.01 + 1.76i)11-s + (−0.523 + 0.906i)12-s + (3.34 + 0.0400i)14-s − 7.62·15-s + (1.52 − 2.63i)16-s + (−1.99 − 3.46i)17-s + (2.46 + 4.27i)18-s + ⋯
L(s)  = 1  + (0.447 − 0.775i)2-s + (0.758 + 1.31i)3-s + (0.0995 + 0.172i)4-s + (−0.649 + 1.12i)5-s + 1.35·6-s + (0.489 + 0.871i)7-s + 1.07·8-s + (−0.650 + 1.12i)9-s + (0.580 + 1.00i)10-s + (0.307 + 0.531i)11-s + (−0.151 + 0.261i)12-s + (0.894 + 0.0107i)14-s − 1.96·15-s + (0.380 − 0.659i)16-s + (−0.484 − 0.839i)17-s + (0.582 + 1.00i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0513 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0513 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.0513 - 0.998i$
Analytic conductor: \(9.44630\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (170, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :1/2),\ -0.0513 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.879608542\)
\(L(\frac12)\) \(\approx\) \(2.879608542\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.29 - 2.30i)T \)
13 \( 1 \)
good2 \( 1 + (-0.632 + 1.09i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.31 - 2.27i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.45 - 2.51i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.01 - 1.76i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.99 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.48 + 6.02i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.313 + 0.543i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.09T + 29T^{2} \)
31 \( 1 + (5.21 + 9.03i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.54 - 2.67i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.521T + 41T^{2} \)
43 \( 1 - 0.329T + 43T^{2} \)
47 \( 1 + (-5.27 + 9.13i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.55 + 6.16i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.01 - 1.76i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.20 - 2.07i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.34 - 12.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.60T + 71T^{2} \)
73 \( 1 + (-1.48 - 2.57i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.38 + 7.58i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12.8T + 83T^{2} \)
89 \( 1 + (1.34 - 2.32i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.32T + 97T^{2} \)
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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

−10.04905902152423586562456881086, −9.329004783249085351373695015247, −8.544453781937748106285746655588, −7.52233242898347853603450285669, −6.90590864697590532003273779603, −5.24601987594226285615840186975, −4.46237112823602587210143850068, −3.70243396208322382864115656749, −2.84468637507219491462951446611, −2.33080952523630176432825047521, 1.10190866919114607791222693522, 1.66794665270985384190261762004, 3.53017543353884194571818430951, 4.41950386254841148427017012609, 5.42051698488800254333919754071, 6.37502112387341545178616589416, 7.20596332557231651325284303206, 7.85699072927449188737024407749, 8.272078103546286233966402443894, 9.130761280563739446430484355370

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