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Properties

Label 2-1183-7.2-c1-0-75
Degree $2$
Conductor $1183$
Sign $-0.761 + 0.647i$
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  + (0.952 − 1.65i)2-s + (−0.214 − 0.371i)3-s + (−0.815 − 1.41i)4-s + (0.736 − 1.27i)5-s − 0.816·6-s + (1.04 − 2.43i)7-s + 0.702·8-s + (1.40 − 2.43i)9-s + (−1.40 − 2.43i)10-s + (2.19 + 3.80i)11-s + (−0.349 + 0.605i)12-s + (−3.01 − 4.03i)14-s − 0.631·15-s + (2.30 − 3.98i)16-s + (0.601 + 1.04i)17-s + (−2.68 − 4.64i)18-s + ⋯
L(s)  = 1  + (0.673 − 1.16i)2-s + (−0.123 − 0.214i)3-s + (−0.407 − 0.706i)4-s + (0.329 − 0.570i)5-s − 0.333·6-s + (0.394 − 0.918i)7-s + 0.248·8-s + (0.469 − 0.813i)9-s + (−0.443 − 0.768i)10-s + (0.662 + 1.14i)11-s + (−0.100 + 0.174i)12-s + (−0.806 − 1.07i)14-s − 0.162·15-s + (0.575 − 0.996i)16-s + (0.145 + 0.252i)17-s + (−0.632 − 1.09i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.761 + 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.761 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.761 + 0.647i$
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: \((2,\ 1183,\ (\ :1/2),\ -0.761 + 0.647i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.821320928\)
\(L(\frac12)\) \(\approx\) \(2.821320928\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.04 + 2.43i)T \)
13 \( 1 \)
good2 \( 1 + (-0.952 + 1.65i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.214 + 0.371i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.736 + 1.27i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.19 - 3.80i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.601 - 1.04i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.62 - 2.80i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.21 + 3.84i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.167T + 29T^{2} \)
31 \( 1 + (2.62 + 4.54i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.52 - 6.10i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.16T + 41T^{2} \)
43 \( 1 - 0.0227T + 43T^{2} \)
47 \( 1 + (5.84 - 10.1i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.0708 - 0.122i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.67 - 4.62i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.77 - 9.99i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.06 + 3.58i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.96T + 71T^{2} \)
73 \( 1 + (7.62 + 13.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.387 - 0.670i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 16.0T + 83T^{2} \)
89 \( 1 + (3.27 - 5.67i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.49T + 97T^{2} \)
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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

−9.783458354729268254466092142708, −8.908538778888163132825346753823, −7.64819088098367994957881379727, −6.99706769659856035134307265859, −5.92668621960965688248735553917, −4.53923295192969277878931286280, −4.36041086220649062948215335704, −3.25774534370430207810459016775, −1.76556814732911963227987015914, −1.16527599217888866878367137103, 1.79404808333237189492533285548, 3.13679713754667639444242213842, 4.34277588019880758428402340407, 5.27535457738087570283951502554, 5.76269731836862240742787120711, 6.67739250679684577141139869410, 7.33158265044426590089189246084, 8.347687741039762263379041154997, 8.974452974591164154297611753322, 10.12141379728507091705160394333

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