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Properties

Label 2-1197-7.4-c1-0-38
Degree $2$
Conductor $1197$
Sign $0.160 + 0.987i$
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  + (−0.198 − 0.343i)2-s + (0.921 − 1.59i)4-s + (−0.421 − 0.729i)5-s + (1.79 + 1.93i)7-s − 1.52·8-s + (−0.167 + 0.289i)10-s + (−0.0179 + 0.0311i)11-s + 1.96·13-s + (0.308 − 1.00i)14-s + (−1.54 − 2.66i)16-s + (1.62 − 2.81i)17-s + (−0.5 − 0.866i)19-s − 1.55·20-s + 0.0142·22-s + (0.641 + 1.11i)23-s + ⋯
L(s)  = 1  + (−0.140 − 0.242i)2-s + (0.460 − 0.798i)4-s + (−0.188 − 0.326i)5-s + (0.680 + 0.732i)7-s − 0.538·8-s + (−0.0528 + 0.0914i)10-s + (−0.00542 + 0.00939i)11-s + 0.544·13-s + (0.0825 − 0.267i)14-s + (−0.385 − 0.667i)16-s + (0.394 − 0.683i)17-s + (−0.114 − 0.198i)19-s − 0.347·20-s + 0.00304·22-s + (0.133 + 0.231i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.160 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1197 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1197\)    =    \(3^{2} \cdot 7 \cdot 19\)
Sign: $0.160 + 0.987i$
Analytic conductor: \(9.55809\)
Root analytic conductor: \(3.09161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1197} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1197,\ (\ :1/2),\ 0.160 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.751126676\)
\(L(\frac12)\) \(\approx\) \(1.751126676\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.79 - 1.93i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.198 + 0.343i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.421 + 0.729i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.0179 - 0.0311i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.96T + 13T^{2} \)
17 \( 1 + (-1.62 + 2.81i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.641 - 1.11i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.06T + 29T^{2} \)
31 \( 1 + (-1.80 + 3.13i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.08 - 1.87i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.130T + 41T^{2} \)
43 \( 1 + 4.50T + 43T^{2} \)
47 \( 1 + (3.99 + 6.91i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.49 + 2.58i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.48 + 11.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.29 + 2.24i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.25 - 3.91i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.64T + 71T^{2} \)
73 \( 1 + (3.10 - 5.38i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.18 + 8.98i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 + (-4.18 - 7.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.55T + 97T^{2} \)
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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

−9.678667052982091884198758788151, −8.708110989356488217787861879319, −8.185868385007104023910041615823, −6.98240567335707374436345031980, −6.17204104241774861207100193135, −5.28238842850349982906706244022, −4.60498619575357231771981373394, −3.07444950948530399163508323284, −2.06258174042419722305670888165, −0.872789561491001490208104310827, 1.40153080155898757198526106432, 2.86603946739434539698498463499, 3.74297490341473224060795413608, 4.64783590275485410726920035376, 5.95171823294247606169313931167, 6.81395988814591184563572545065, 7.47255364275255378532322455421, 8.256239013828553398691573958143, 8.767435606418154478344212568951, 10.09818848169590297471477908700

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