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Properties

Label 2-1183-7.2-c1-0-50
Degree $2$
Conductor $1183$
Sign $-0.386 - 0.922i$
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  + (−1.30 + 2.26i)2-s + (1.11 + 1.93i)3-s + (−2.42 − 4.20i)4-s + (1.11 − 1.93i)5-s − 5.85·6-s + (2 + 1.73i)7-s + 7.47·8-s + (−1 + 1.73i)9-s + (2.92 + 5.06i)10-s + (−1.5 − 2.59i)11-s + (5.42 − 9.39i)12-s + (−6.54 + 2.26i)14-s + 5.00·15-s + (−4.92 + 8.53i)16-s + (−0.736 − 1.27i)17-s + (−2.61 − 4.53i)18-s + ⋯
L(s)  = 1  + (−0.925 + 1.60i)2-s + (0.645 + 1.11i)3-s + (−1.21 − 2.10i)4-s + (0.499 − 0.866i)5-s − 2.38·6-s + (0.755 + 0.654i)7-s + 2.64·8-s + (−0.333 + 0.577i)9-s + (0.925 + 1.60i)10-s + (−0.452 − 0.783i)11-s + (1.56 − 2.71i)12-s + (−1.74 + 0.605i)14-s + 1.29·15-s + (−1.23 + 2.13i)16-s + (−0.178 − 0.309i)17-s + (−0.617 − 1.06i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.386 - 0.922i$
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: \((2,\ 1183,\ (\ :1/2),\ -0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.420744104\)
\(L(\frac12)\) \(\approx\) \(1.420744104\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2 - 1.73i)T \)
13 \( 1 \)
good2 \( 1 + (1.30 - 2.26i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.11 - 1.93i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.11 + 1.93i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.736 + 1.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.5 + 2.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.11 + 7.13i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 + (-2.5 - 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.35 + 4.07i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.47T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (3.73 - 6.47i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.73 - 6.47i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.736 + 1.27i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.5 - 2.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.5 + 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.94T + 71T^{2} \)
73 \( 1 + (1.35 + 2.34i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.35 + 2.34i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-1.11 + 1.93i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.41T + 97T^{2} \)
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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

−9.498088950410223856921703022114, −8.936834686423981543617691074049, −8.606587409345371044129846908759, −7.942309852513652209077249571904, −6.75286636278374088542904287927, −5.79503195350725474188478093557, −4.90077805720536869481317503158, −4.67872993610053056427934158397, −2.79488112877270414127612213735, −0.988314660888264588742939175370, 1.17834069280827079155607748349, 1.96460054055032530093593554931, 2.70022235914273561840142761034, 3.66496200295194288997885792236, 4.91784962875383524269049550069, 6.64868455019575157540133013813, 7.47060041285679429956367221736, 7.962305996115288866199064800554, 8.696698642927906473572804714167, 9.921838373570165724073668412923

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