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Properties

Label 2-1183-7.4-c1-0-81
Degree $2$
Conductor $1183$
Sign $-0.243 - 0.969i$
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  + (−1.03 − 1.79i)2-s + (1.27 − 2.20i)3-s + (−1.13 + 1.97i)4-s + (−0.427 − 0.741i)5-s − 5.26·6-s + (0.562 − 2.58i)7-s + 0.572·8-s + (−1.73 − 3.00i)9-s + (−0.884 + 1.53i)10-s + (3.02 − 5.23i)11-s + (2.89 + 5.01i)12-s + (−5.21 + 1.66i)14-s − 2.17·15-s + (1.68 + 2.91i)16-s + (−2.35 + 4.07i)17-s + (−3.59 + 6.21i)18-s + ⋯
L(s)  = 1  + (−0.731 − 1.26i)2-s + (0.734 − 1.27i)3-s + (−0.569 + 0.985i)4-s + (−0.191 − 0.331i)5-s − 2.14·6-s + (0.212 − 0.977i)7-s + 0.202·8-s + (−0.578 − 1.00i)9-s + (−0.279 + 0.484i)10-s + (0.911 − 1.57i)11-s + (0.836 + 1.44i)12-s + (−1.39 + 0.445i)14-s − 0.562·15-s + (0.421 + 0.729i)16-s + (−0.570 + 0.987i)17-s + (−0.846 + 1.46i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.243 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.243 - 0.969i$
Analytic conductor: \(9.44630\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :1/2),\ -0.243 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.269119413\)
\(L(\frac12)\) \(\approx\) \(1.269119413\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.562 + 2.58i)T \)
13 \( 1 \)
good2 \( 1 + (1.03 + 1.79i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.27 + 2.20i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.427 + 0.741i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.02 + 5.23i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.35 - 4.07i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.48 + 2.57i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.62 + 2.81i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.50T + 29T^{2} \)
31 \( 1 + (0.970 - 1.68i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.23 - 7.34i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.47T + 41T^{2} \)
43 \( 1 - 3.13T + 43T^{2} \)
47 \( 1 + (-1.52 - 2.63i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.98 + 6.89i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.32 - 5.76i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.60 - 9.71i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.87 - 8.43i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.35T + 71T^{2} \)
73 \( 1 + (1.78 - 3.09i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.405 - 0.702i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 17.0T + 83T^{2} \)
89 \( 1 + (7.59 + 13.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 15.2T + 97T^{2} \)
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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

−8.835890235984563759828280244517, −8.576760446289759606207142010635, −8.013936478208213980483071066160, −6.76494728154233324653443900341, −6.23566410226330577918809880702, −4.35832833306136626734153008102, −3.44335496173857931101940558422, −2.50190586948100868094432059104, −1.34275415558837543514544824218, −0.69962585981502450985978099889, 2.19204123097666026482000039009, 3.41577443013819795440591699025, 4.51732086144241238268494172259, 5.25693608616076705021207909254, 6.39261022561668980407899713375, 7.18065473460302059520601314912, 7.991984810881573161797512136040, 8.849493158030816189698946528257, 9.423308750537074832492055115198, 9.690980082125321352231432761505

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