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Properties

Label 2-1197-7.2-c1-0-45
Degree $2$
Conductor $1197$
Sign $-0.0547 + 0.998i$
Analytic cond. $9.55809$
Root an. cond. $3.09161$
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  + (−1.38 + 2.40i)2-s + (−2.84 − 4.92i)4-s + (−0.602 + 1.04i)5-s + (1.58 − 2.11i)7-s + 10.2·8-s + (−1.67 − 2.89i)10-s + (−2.15 − 3.73i)11-s − 1.16·13-s + (2.89 + 6.74i)14-s + (−8.48 + 14.6i)16-s + (1.25 + 2.17i)17-s + (0.5 − 0.866i)19-s + 6.85·20-s + 11.9·22-s + (−0.277 + 0.480i)23-s + ⋯
L(s)  = 1  + (−0.980 + 1.69i)2-s + (−1.42 − 2.46i)4-s + (−0.269 + 0.466i)5-s + (0.598 − 0.801i)7-s + 3.61·8-s + (−0.528 − 0.914i)10-s + (−0.650 − 1.12i)11-s − 0.323·13-s + (0.773 + 1.80i)14-s + (−2.12 + 3.67i)16-s + (0.304 + 0.526i)17-s + (0.114 − 0.198i)19-s + 1.53·20-s + 2.55·22-s + (−0.0577 + 0.100i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0547 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1197 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0547 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1197\)    =    \(3^{2} \cdot 7 \cdot 19\)
Sign: $-0.0547 + 0.998i$
Analytic conductor: \(9.55809\)
Root analytic conductor: \(3.09161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1197} (856, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1197,\ (\ :1/2),\ -0.0547 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.02030331956\)
\(L(\frac12)\) \(\approx\) \(0.02030331956\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-1.58 + 2.11i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (1.38 - 2.40i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.602 - 1.04i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.15 + 3.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.16T + 13T^{2} \)
17 \( 1 + (-1.25 - 2.17i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (0.277 - 0.480i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.23T + 29T^{2} \)
31 \( 1 + (2.57 + 4.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.56 - 9.63i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.03T + 41T^{2} \)
43 \( 1 + 1.10T + 43T^{2} \)
47 \( 1 + (3.82 - 6.62i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.268 - 0.464i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.33 + 10.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.210 + 0.363i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.06 + 3.57i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.92T + 71T^{2} \)
73 \( 1 + (-6.36 - 11.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.46 - 5.99i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 14.0T + 83T^{2} \)
89 \( 1 + (2.68 - 4.65i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.2T + 97T^{2} \)
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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

−9.340037010411891076671821380372, −8.311073158112710728606102949161, −7.947917172535107174631150668286, −7.19076741759499631489659966100, −6.47988802633788791150542012210, −5.50843343252361861053506290701, −4.83840202868923751776948677504, −3.55329252646516663543898855329, −1.45126789103971354413539802841, −0.01273691377471194645553042023, 1.64606376918245598094379179997, 2.37577287816942034119535985964, 3.48508704675021394943168248338, 4.63897835353979747851143953435, 5.21549020007988975550559507820, 7.25849828667781117141749225822, 7.80742486602531860876213580830, 8.743911867590212521537569156507, 9.134285242959560307245965625990, 10.11420717492708449477812345177

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