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Properties

Label 2-1183-91.69-c0-0-3
Degree $2$
Conductor $1183$
Sign $-0.820 + 0.571i$
Analytic cond. $0.590393$
Root an. cond. $0.768370$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.56 − 0.900i)2-s + (1.12 + 1.94i)4-s + (0.866 − 0.5i)7-s − 2.24i·8-s + (−0.5 − 0.866i)9-s + (−1.07 − 0.623i)11-s − 1.80·14-s + (−0.900 + 1.56i)16-s + 1.80i·18-s + (1.12 + 1.94i)22-s + (−0.222 + 0.385i)23-s − 25-s + (1.94 + 1.12i)28-s + (0.900 − 1.56i)29-s + (0.866 − 0.500i)32-s + ⋯
L(s)  = 1  + (−1.56 − 0.900i)2-s + (1.12 + 1.94i)4-s + (0.866 − 0.5i)7-s − 2.24i·8-s + (−0.5 − 0.866i)9-s + (−1.07 − 0.623i)11-s − 1.80·14-s + (−0.900 + 1.56i)16-s + 1.80i·18-s + (1.12 + 1.94i)22-s + (−0.222 + 0.385i)23-s − 25-s + (1.94 + 1.12i)28-s + (0.900 − 1.56i)29-s + (0.866 − 0.500i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.820 + 0.571i$
Analytic conductor: \(0.590393\)
Root analytic conductor: \(0.768370\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :0),\ -0.820 + 0.571i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4002479326\)
\(L(\frac12)\) \(\approx\) \(0.4002479326\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 \)
good2 \( 1 + (1.56 + 0.900i)T + (0.5 + 0.866i)T^{2} \)
3 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + T^{2} \)
11 \( 1 + (1.07 + 0.623i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.222 - 0.385i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.385 - 0.222i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.900 + 1.56i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 0.445T + T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.385 + 0.222i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.385 + 0.222i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - 1.24T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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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

−9.751636533691206087764500053174, −8.853075638554011130055025379505, −8.077391901352949553936789716160, −7.77079176100598326732748140558, −6.59822290416715270379364244511, −5.45590903721192736572324888725, −4.01583947347853509750353650154, −3.01676408091180027749160511826, −1.97934743768011722467163890560, −0.57910912862630975142247561335, 1.66282887674442046751353981201, 2.60924768160890316813221919230, 4.79358548674162351781482125753, 5.39676528070459447614334886539, 6.32982585457709219179386836996, 7.38192144174170516676324026563, 8.015521058839194258037550743429, 8.392175369913507173321192521221, 9.306234582399032999394841114414, 10.16121975633565484445041734799

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