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Properties

Label 2-1183-7.4-c1-0-19
Degree $2$
Conductor $1183$
Sign $-0.550 + 0.834i$
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  + (−1.21 − 2.10i)2-s + (−0.979 + 1.69i)3-s + (−1.96 + 3.40i)4-s + (−2.07 − 3.59i)5-s + 4.76·6-s + (−2.28 + 1.33i)7-s + 4.69·8-s + (−0.416 − 0.722i)9-s + (−5.05 + 8.74i)10-s + (−0.336 + 0.582i)11-s + (−3.84 − 6.66i)12-s + (5.59 + 3.19i)14-s + 8.12·15-s + (−1.78 − 3.09i)16-s + (0.0504 − 0.0874i)17-s + (−1.01 + 1.75i)18-s + ⋯
L(s)  = 1  + (−0.860 − 1.49i)2-s + (−0.565 + 0.979i)3-s + (−0.981 + 1.70i)4-s + (−0.927 − 1.60i)5-s + 1.94·6-s + (−0.863 + 0.503i)7-s + 1.65·8-s + (−0.138 − 0.240i)9-s + (−1.59 + 2.76i)10-s + (−0.101 + 0.175i)11-s + (−1.11 − 1.92i)12-s + (1.49 + 0.854i)14-s + 2.09·15-s + (−0.446 − 0.773i)16-s + (0.0122 − 0.0211i)17-s + (−0.239 + 0.414i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.550 + 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.550 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.550 + 0.834i$
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: \((2,\ 1183,\ (\ :1/2),\ -0.550 + 0.834i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3254205830\)
\(L(\frac12)\) \(\approx\) \(0.3254205830\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.28 - 1.33i)T \)
13 \( 1 \)
good2 \( 1 + (1.21 + 2.10i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.979 - 1.69i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (2.07 + 3.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.336 - 0.582i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.0504 + 0.0874i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.72 - 4.71i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.231 + 0.400i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.30T + 29T^{2} \)
31 \( 1 + (3.61 - 6.26i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.62 + 4.54i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.44T + 41T^{2} \)
43 \( 1 - 4.83T + 43T^{2} \)
47 \( 1 + (-1.41 - 2.45i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.17 + 12.4i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.208 - 0.361i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.768 + 1.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.94 + 5.09i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.0T + 71T^{2} \)
73 \( 1 + (-3.19 + 5.54i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.143 + 0.249i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.43T + 83T^{2} \)
89 \( 1 + (5.02 + 8.70i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.84T + 97T^{2} \)
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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

−9.639335777694291892679940901692, −8.955194800374602024672327292357, −8.397529897097840675756398549064, −7.44733967842482608425294859008, −5.68462019833700273626834317349, −4.90832933421610327348190800879, −3.95678812329928560251728061076, −3.37899039110363918918105730485, −1.73850407812394181316531483299, −0.35441075093533553163489643274, 0.68724285241478148535467494262, 2.82872917393531415548410717941, 4.01130311955753384911896430123, 5.60041971830756538428180830350, 6.41506082054531249864660175794, 6.87296608872181616863868591112, 7.37458004524854288780879776769, 7.85690109395364467626688540650, 9.049487004953381797383955240884, 9.945740712822727716619395967885

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