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Properties

Label 2-1197-7.2-c1-0-14
Degree $2$
Conductor $1197$
Sign $-0.519 - 0.854i$
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  + (0.323 − 0.559i)2-s + (0.790 + 1.36i)4-s + (−2.12 + 3.68i)5-s + (−0.107 + 2.64i)7-s + 2.31·8-s + (1.37 + 2.38i)10-s + (−0.155 − 0.269i)11-s + 5.14·13-s + (1.44 + 0.914i)14-s + (−0.833 + 1.44i)16-s + (0.513 + 0.889i)17-s + (0.5 − 0.866i)19-s − 6.72·20-s − 0.201·22-s + (1.32 − 2.28i)23-s + ⋯
L(s)  = 1  + (0.228 − 0.395i)2-s + (0.395 + 0.684i)4-s + (−0.951 + 1.64i)5-s + (−0.0404 + 0.999i)7-s + 0.818·8-s + (0.434 + 0.753i)10-s + (−0.0469 − 0.0813i)11-s + 1.42·13-s + (0.386 + 0.244i)14-s + (−0.208 + 0.360i)16-s + (0.124 + 0.215i)17-s + (0.114 − 0.198i)19-s − 1.50·20-s − 0.0429·22-s + (0.275 − 0.477i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.519 - 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1197 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.519 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1197\)    =    \(3^{2} \cdot 7 \cdot 19\)
Sign: $-0.519 - 0.854i$
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: \((2,\ 1197,\ (\ :1/2),\ -0.519 - 0.854i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.689887939\)
\(L(\frac12)\) \(\approx\) \(1.689887939\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.107 - 2.64i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.323 + 0.559i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (2.12 - 3.68i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.155 + 0.269i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.14T + 13T^{2} \)
17 \( 1 + (-0.513 - 0.889i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.32 + 2.28i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.32T + 29T^{2} \)
31 \( 1 + (-2.39 - 4.15i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.12 - 5.41i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 8.94T + 41T^{2} \)
43 \( 1 + 6.50T + 43T^{2} \)
47 \( 1 + (4.33 - 7.50i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.251 - 0.435i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.14 + 7.18i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.74 + 4.76i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.46 + 6.00i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.94T + 71T^{2} \)
73 \( 1 + (7.46 + 12.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.45 + 12.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.4T + 83T^{2} \)
89 \( 1 + (0.0485 - 0.0841i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.4T + 97T^{2} \)
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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

−10.49322265336783184478679025507, −9.118982865149381296929667521017, −8.168464389684297442652318531484, −7.67747750554401571364531704953, −6.60243469905335405904679307677, −6.19241268429584230517320918343, −4.61676018546977432967019703309, −3.33435541408974245492206546777, −3.24897757600588518789758929923, −2.01741432895820725030274867793, 0.70877483840739164443957090884, 1.54522113070800960677317053260, 3.70135663169549824498499858282, 4.29519055277961946441152491802, 5.23545629418460815902913016948, 5.94896259844737777973594975392, 7.14669917451895310424554065205, 7.73046461469894116989551195183, 8.562586117460760740677030644961, 9.384753543102443917436995159835

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