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Properties

Label 2-1183-7.4-c1-0-88
Degree $2$
Conductor $1183$
Sign $-0.905 - 0.423i$
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  + (−0.0904 − 0.156i)2-s + (0.913 − 1.58i)3-s + (0.983 − 1.70i)4-s + (−1.34 − 2.32i)5-s − 0.330·6-s + (−1.64 − 2.06i)7-s − 0.717·8-s + (−0.167 − 0.289i)9-s + (−0.242 + 0.420i)10-s + (−1.34 + 2.33i)11-s + (−1.79 − 3.11i)12-s + (−0.174 + 0.445i)14-s − 4.90·15-s + (−1.90 − 3.29i)16-s + (−2.38 + 4.12i)17-s + (−0.0302 + 0.0523i)18-s + ⋯
L(s)  = 1  + (−0.0639 − 0.110i)2-s + (0.527 − 0.913i)3-s + (0.491 − 0.851i)4-s + (−0.600 − 1.04i)5-s − 0.134·6-s + (−0.623 − 0.781i)7-s − 0.253·8-s + (−0.0557 − 0.0965i)9-s + (−0.0768 + 0.133i)10-s + (−0.406 + 0.703i)11-s + (−0.518 − 0.898i)12-s + (−0.0467 + 0.119i)14-s − 1.26·15-s + (−0.475 − 0.823i)16-s + (−0.577 + 1.00i)17-s + (−0.00712 + 0.0123i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 - 0.423i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.905 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.905 - 0.423i$
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: \((2,\ 1183,\ (\ :1/2),\ -0.905 - 0.423i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.276347453\)
\(L(\frac12)\) \(\approx\) \(1.276347453\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.64 + 2.06i)T \)
13 \( 1 \)
good2 \( 1 + (0.0904 + 0.156i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.913 + 1.58i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.34 + 2.32i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.34 - 2.33i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.38 - 4.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0942 + 0.163i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.19 + 3.80i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.08T + 29T^{2} \)
31 \( 1 + (-1.84 + 3.20i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.97 + 6.88i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.42T + 41T^{2} \)
43 \( 1 + 8.01T + 43T^{2} \)
47 \( 1 + (-0.924 - 1.60i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.53 + 6.12i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.79 - 6.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.205 - 0.356i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.70 - 9.87i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.34T + 71T^{2} \)
73 \( 1 + (-7.10 + 12.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.55 + 7.89i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16.5T + 83T^{2} \)
89 \( 1 + (-2.94 - 5.10i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.451T + 97T^{2} \)
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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

−9.212571429710388388128386768687, −8.375759521217285847693934973022, −7.66588938243621531411869234699, −6.86458261242331503984572675120, −6.19059961733227040133618275054, −4.88270154799982469756992103839, −4.13749639902624258749290878235, −2.61524830507706436060570375621, −1.62543369164855683971409800757, −0.50326324523832122232639150584, 2.63976043858464929526997602021, 3.11279213715084068936350404629, 3.75287181142715863564795029635, 4.97806489102892406606661679735, 6.36781925671143280450990973491, 6.86747187387217391534716831518, 7.894858764281435546683358476914, 8.602820976383359245178914835675, 9.325912889295127246859545191168, 10.22161195558536304893702931885

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