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Properties

Label 2-1183-7.2-c1-0-52
Degree $2$
Conductor $1183$
Sign $0.989 + 0.145i$
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  + (−0.545 + 0.945i)2-s + (−0.697 − 1.20i)3-s + (0.403 + 0.699i)4-s + (0.813 − 1.40i)5-s + 1.52·6-s + (2.51 − 0.817i)7-s − 3.06·8-s + (0.526 − 0.911i)9-s + (0.888 + 1.53i)10-s + (0.646 + 1.12i)11-s + (0.563 − 0.975i)12-s + (−0.600 + 2.82i)14-s − 2.26·15-s + (0.866 − 1.50i)16-s + (2.70 + 4.68i)17-s + (0.574 + 0.995i)18-s + ⋯
L(s)  = 1  + (−0.386 + 0.668i)2-s + (−0.402 − 0.697i)3-s + (0.201 + 0.349i)4-s + (0.363 − 0.629i)5-s + 0.622·6-s + (0.951 − 0.309i)7-s − 1.08·8-s + (0.175 − 0.303i)9-s + (0.280 + 0.486i)10-s + (0.195 + 0.337i)11-s + (0.162 − 0.281i)12-s + (−0.160 + 0.755i)14-s − 0.586·15-s + (0.216 − 0.375i)16-s + (0.656 + 1.13i)17-s + (0.135 + 0.234i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.145i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.989 + 0.145i$
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: \((2,\ 1183,\ (\ :1/2),\ 0.989 + 0.145i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.499990394\)
\(L(\frac12)\) \(\approx\) \(1.499990394\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.51 + 0.817i)T \)
13 \( 1 \)
good2 \( 1 + (0.545 - 0.945i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.697 + 1.20i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.813 + 1.40i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.646 - 1.12i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.70 - 4.68i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.755 + 1.30i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.32 + 2.28i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.81T + 29T^{2} \)
31 \( 1 + (3.64 + 6.30i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.47 + 6.02i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.09T + 41T^{2} \)
43 \( 1 - 11.1T + 43T^{2} \)
47 \( 1 + (3.58 - 6.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.33 + 4.04i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.386 + 0.670i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.37 - 7.57i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.18 + 5.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 + (3.60 + 6.24i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.88 + 10.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.42T + 83T^{2} \)
89 \( 1 + (-0.833 + 1.44i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.4T + 97T^{2} \)
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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

−9.421243489426143895251855507725, −8.808189712712310022052337669856, −7.83100842311482537112418372902, −7.46360647993130181732000691002, −6.45454828655831768098757136480, −5.85076779960809742061798044951, −4.77065940932360376720580795879, −3.65436556821323127284734939426, −2.05090897808161313620828567569, −0.932335810757812296275792347703, 1.21960444227231288825978046579, 2.38550978657888213863865991480, 3.34645791456553755334384610974, 4.82099774790752135166678678067, 5.37993339973484056490471316796, 6.31994161506721318559623967870, 7.31062067112113098128613304037, 8.375066576898917722760102717780, 9.294303090096873402054408188280, 10.03107094629136001744456225814

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