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Properties

Label 2-1183-7.4-c1-0-1
Degree $2$
Conductor $1183$
Sign $0.458 - 0.888i$
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  + (−1.29 − 2.24i)2-s + (−0.259 + 0.449i)3-s + (−2.35 + 4.07i)4-s + (−0.806 − 1.39i)5-s + 1.34·6-s + (2.13 − 1.56i)7-s + 6.99·8-s + (1.36 + 2.36i)9-s + (−2.08 + 3.61i)10-s + (−1.35 + 2.34i)11-s + (−1.21 − 2.11i)12-s + (−6.27 − 2.74i)14-s + 0.835·15-s + (−4.34 − 7.53i)16-s + (−1.56 + 2.70i)17-s + (3.53 − 6.12i)18-s + ⋯
L(s)  = 1  + (−0.915 − 1.58i)2-s + (−0.149 + 0.259i)3-s + (−1.17 + 2.03i)4-s + (−0.360 − 0.624i)5-s + 0.547·6-s + (0.805 − 0.592i)7-s + 2.47·8-s + (0.455 + 0.788i)9-s + (−0.659 + 1.14i)10-s + (−0.407 + 0.706i)11-s + (−0.351 − 0.609i)12-s + (−1.67 − 0.734i)14-s + 0.215·15-s + (−1.08 − 1.88i)16-s + (−0.379 + 0.656i)17-s + (0.833 − 1.44i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.458 - 0.888i$
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: \((2,\ 1183,\ (\ :1/2),\ 0.458 - 0.888i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1653313030\)
\(L(\frac12)\) \(\approx\) \(0.1653313030\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.13 + 1.56i)T \)
13 \( 1 \)
good2 \( 1 + (1.29 + 2.24i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.259 - 0.449i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.806 + 1.39i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.35 - 2.34i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.56 - 2.70i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.84 + 3.18i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.993 + 1.71i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.37T + 29T^{2} \)
31 \( 1 + (5.23 - 9.07i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.97 + 5.15i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.70T + 41T^{2} \)
43 \( 1 + 3.35T + 43T^{2} \)
47 \( 1 + (0.527 + 0.913i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.63 - 6.29i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.71 - 9.89i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.46 - 2.53i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.79 + 11.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.35T + 71T^{2} \)
73 \( 1 + (4.55 - 7.88i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.10 - 5.37i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.69T + 83T^{2} \)
89 \( 1 + (0.879 + 1.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 15.4T + 97T^{2} \)
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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

−10.31099834885195672029362702627, −9.151583104819648020014668190187, −8.541872643141242388046470951506, −7.81994684311129820518711732817, −7.08076318099517390644831547401, −5.06367220733164010475510398733, −4.50550141808332337657044323109, −3.70898421566403296955834350136, −2.21982623114031360316874916545, −1.46409758925476026826130902858, 0.10241837375159672444860859081, 1.72791696579539042033816644611, 3.56529130194991711275072947798, 4.92560343988383217774799893734, 5.74685531207941753577139118969, 6.44083878507991321580892630760, 7.27068006747727694198966341378, 7.85088714810851947158857374895, 8.586213255926925497192870327609, 9.350874575340785133986912321366

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