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Properties

Label 2-1183-7.2-c1-0-60
Degree $2$
Conductor $1183$
Sign $0.722 + 0.691i$
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  + (0.294 − 0.509i)2-s + (−0.386 − 0.670i)3-s + (0.826 + 1.43i)4-s + (1.34 − 2.33i)5-s − 0.455·6-s + (2.52 + 0.791i)7-s + 2.15·8-s + (1.20 − 2.07i)9-s + (−0.793 − 1.37i)10-s + (1.34 + 2.33i)11-s + (0.639 − 1.10i)12-s + (1.14 − 1.05i)14-s − 2.08·15-s + (−1.02 + 1.76i)16-s + (0.237 + 0.411i)17-s + (−0.706 − 1.22i)18-s + ⋯
L(s)  = 1  + (0.208 − 0.360i)2-s + (−0.223 − 0.386i)3-s + (0.413 + 0.716i)4-s + (0.602 − 1.04i)5-s − 0.185·6-s + (0.954 + 0.298i)7-s + 0.760·8-s + (0.400 − 0.693i)9-s + (−0.250 − 0.434i)10-s + (0.405 + 0.703i)11-s + (0.184 − 0.319i)12-s + (0.306 − 0.281i)14-s − 0.538·15-s + (−0.255 + 0.442i)16-s + (0.0576 + 0.0998i)17-s + (−0.166 − 0.288i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.722 + 0.691i$
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: \((2,\ 1183,\ (\ :1/2),\ 0.722 + 0.691i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.555914200\)
\(L(\frac12)\) \(\approx\) \(2.555914200\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.52 - 0.791i)T \)
13 \( 1 \)
good2 \( 1 + (-0.294 + 0.509i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.386 + 0.670i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.34 + 2.33i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.34 - 2.33i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.237 - 0.411i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.980 - 1.69i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.86 + 4.95i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.40T + 29T^{2} \)
31 \( 1 + (-1.65 - 2.86i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.44 - 9.42i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 11.6T + 41T^{2} \)
43 \( 1 + 9.43T + 43T^{2} \)
47 \( 1 + (-4.49 + 7.78i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.15 - 3.72i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.96 + 8.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.98 + 8.63i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.70 - 2.95i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.27T + 71T^{2} \)
73 \( 1 + (-2.10 - 3.65i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.39 + 12.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.99T + 83T^{2} \)
89 \( 1 + (1.83 - 3.18i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 17.3T + 97T^{2} \)
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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

−9.705361371989961392224721467285, −8.588912556291380204061125132654, −8.301513070744734542350587456544, −7.02546261438228087777464779538, −6.51307631063149522736796560269, −5.13922307047883683761509876924, −4.60296520987971882389052493697, −3.45917947056189365952401189738, −1.97515690486649654169931589714, −1.36102761363542305298204629891, 1.43980605582934465690213912279, 2.44903957500171720683051160049, 3.86993383304677143745357761897, 5.03256598872590147793634252307, 5.53284155029899186909274866267, 6.57061907651683071275074659231, 7.15464180221338523739921787731, 8.017761294794633300265184269844, 9.213918623820730434716662995721, 10.17160585905439848586600873637

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