Properties

Label 2-1183-7.4-c1-0-49
Degree $2$
Conductor $1183$
Sign $0.283 - 0.958i$
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  + (0.883 + 1.53i)2-s + (1.16 − 2.01i)3-s + (−0.561 + 0.972i)4-s + (2.03 + 3.52i)5-s + 4.11·6-s + (−1.79 + 1.93i)7-s + 1.54·8-s + (−1.20 − 2.09i)9-s + (−3.59 + 6.22i)10-s + (1.45 − 2.51i)11-s + (1.30 + 2.26i)12-s + (−4.55 − 1.03i)14-s + 9.46·15-s + (2.49 + 4.31i)16-s + (1.68 − 2.92i)17-s + (2.13 − 3.70i)18-s + ⋯
L(s)  = 1  + (0.624 + 1.08i)2-s + (0.672 − 1.16i)3-s + (−0.280 + 0.486i)4-s + (0.909 + 1.57i)5-s + 1.67·6-s + (−0.680 + 0.733i)7-s + 0.547·8-s + (−0.403 − 0.698i)9-s + (−1.13 + 1.96i)10-s + (0.438 − 0.759i)11-s + (0.377 + 0.653i)12-s + (−1.21 − 0.277i)14-s + 2.44·15-s + (0.623 + 1.07i)16-s + (0.409 − 0.709i)17-s + (0.503 − 0.872i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.283 - 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.283 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.283 - 0.958i$
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: \((2,\ 1183,\ (\ :1/2),\ 0.283 - 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.422655312\)
\(L(\frac12)\) \(\approx\) \(3.422655312\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.79 - 1.93i)T \)
13 \( 1 \)
good2 \( 1 + (-0.883 - 1.53i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.16 + 2.01i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.03 - 3.52i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.45 + 2.51i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.68 + 2.92i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.593 + 1.02i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.00373 - 0.00646i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.00T + 29T^{2} \)
31 \( 1 + (0.339 - 0.588i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.29 - 2.24i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.5T + 41T^{2} \)
43 \( 1 - 4.33T + 43T^{2} \)
47 \( 1 + (-5.70 - 9.88i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.04 + 5.28i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.97 + 3.42i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.80 + 3.12i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.838 - 1.45i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.13T + 71T^{2} \)
73 \( 1 + (-5.24 + 9.08i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.72 + 9.90i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.82T + 83T^{2} \)
89 \( 1 + (1.80 + 3.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−9.764639045755265400189216704813, −8.943948650039031238275433825321, −7.86991926735108878624437816115, −7.15644135890466914957523794217, −6.59841903495562556789380997923, −6.07935562190469647112794851733, −5.34635462160274993827150246065, −3.52005519473083866320524392946, −2.74293976009722342143628618122, −1.78163411426351516182663722609, 1.27891612964553447516307824829, 2.31903169402296646889925358048, 3.75300114197514970315379800377, 4.02647556523219886354741080442, 4.91031187988736192082114288921, 5.74010872112730447496668231816, 7.17447219112789078626738765687, 8.374034050740164514291558717316, 9.154433623424354293936976693849, 9.905705215786555183674733597264

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