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Properties

Label 2-1197-7.2-c1-0-11
Degree $2$
Conductor $1197$
Sign $-0.736 - 0.676i$
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  + (−0.250 + 0.434i)2-s + (0.874 + 1.51i)4-s + (0.568 − 0.984i)5-s + (0.630 + 2.56i)7-s − 1.88·8-s + (0.285 + 0.494i)10-s + (1.67 + 2.89i)11-s − 3.73·13-s + (−1.27 − 0.370i)14-s + (−1.27 + 2.21i)16-s + (−0.716 − 1.24i)17-s + (0.5 − 0.866i)19-s + 1.98·20-s − 1.67·22-s + (−2.48 + 4.31i)23-s + ⋯
L(s)  = 1  + (−0.177 + 0.307i)2-s + (0.437 + 0.756i)4-s + (0.254 − 0.440i)5-s + (0.238 + 0.971i)7-s − 0.664·8-s + (0.0902 + 0.156i)10-s + (0.504 + 0.874i)11-s − 1.03·13-s + (−0.340 − 0.0991i)14-s + (−0.319 + 0.552i)16-s + (−0.173 − 0.301i)17-s + (0.114 − 0.198i)19-s + 0.444·20-s − 0.358·22-s + (−0.519 + 0.899i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1197 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1197\)    =    \(3^{2} \cdot 7 \cdot 19\)
Sign: $-0.736 - 0.676i$
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: \((2,\ 1197,\ (\ :1/2),\ -0.736 - 0.676i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.376756971\)
\(L(\frac12)\) \(\approx\) \(1.376756971\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.630 - 2.56i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.250 - 0.434i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-0.568 + 0.984i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.67 - 2.89i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.73T + 13T^{2} \)
17 \( 1 + (0.716 + 1.24i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (2.48 - 4.31i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.87T + 29T^{2} \)
31 \( 1 + (-0.619 - 1.07i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.17 + 5.50i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.84T + 41T^{2} \)
43 \( 1 - 6.52T + 43T^{2} \)
47 \( 1 + (-1.83 + 3.17i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.45 - 11.1i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.746 + 1.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.82 + 3.15i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.97 - 6.88i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.24T + 71T^{2} \)
73 \( 1 + (-2.42 - 4.19i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.860 - 1.48i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 15.2T + 83T^{2} \)
89 \( 1 + (3.46 - 6.00i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 17.6T + 97T^{2} \)
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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

−9.650396034756017480360850482421, −9.252291835659468532608779159536, −8.463818560576291422480016883973, −7.48063454187198645416419687114, −7.01258689087265199422725733697, −5.84783505476106492727741805893, −5.09102008506240048920819646796, −3.99481027067502984914476540733, −2.71382378429198231948002598433, −1.85815460738361584599187284550, 0.59069721109782545474897007936, 1.90768739251185333763340330284, 2.97339106988296830876447073524, 4.16043503021873205912658674276, 5.22499768683704376538697515471, 6.28299531465680747236702325880, 6.76908475613732821206065854353, 7.75624352352403114552137827950, 8.728068354002849542721868718000, 9.761502479495924487067664883941

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