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Properties

Label 2-1197-7.4-c1-0-36
Degree $2$
Conductor $1197$
Sign $-0.266 - 0.963i$
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 + 1.73i)2-s + (−0.999 + 1.73i)4-s + (0.5 + 0.866i)5-s + (2 + 1.73i)7-s + (−0.999 + 1.73i)10-s + (2 − 3.46i)11-s + 4·13-s + (−0.999 + 5.19i)14-s + (1.99 + 3.46i)16-s + (1.5 − 2.59i)17-s + (0.5 + 0.866i)19-s − 1.99·20-s + 7.99·22-s + (−1.5 − 2.59i)23-s + (2 − 3.46i)25-s + (4 + 6.92i)26-s + ⋯
L(s)  = 1  + (0.707 + 1.22i)2-s + (−0.499 + 0.866i)4-s + (0.223 + 0.387i)5-s + (0.755 + 0.654i)7-s + (−0.316 + 0.547i)10-s + (0.603 − 1.04i)11-s + 1.10·13-s + (−0.267 + 1.38i)14-s + (0.499 + 0.866i)16-s + (0.363 − 0.630i)17-s + (0.114 + 0.198i)19-s − 0.447·20-s + 1.70·22-s + (−0.312 − 0.541i)23-s + (0.400 − 0.692i)25-s + (0.784 + 1.35i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1197 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1197\)    =    \(3^{2} \cdot 7 \cdot 19\)
Sign: $-0.266 - 0.963i$
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: \((2,\ 1197,\ (\ :1/2),\ -0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.033904325\)
\(L(\frac12)\) \(\approx\) \(3.033904325\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 1.73i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-1 - 1.73i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 10T + 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3 - 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 7T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6 - 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5 - 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (3 - 5.19i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3T + 83T^{2} \)
89 \( 1 + (7 + 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 12T + 97T^{2} \)
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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

−9.892786872988371826042123126679, −8.738024783280089023136032357369, −8.286427150672961414179379640069, −7.38952075981755010033804826381, −6.36526725542690696131063994600, −5.94572643188098069192041258008, −5.14814043757902004970648201263, −4.13318000854646644873592841197, −3.11674797638828561074673624008, −1.53557872247746337890156215252, 1.36145542621407559053202370703, 1.85822919228122230212410417769, 3.49398011524995401196443638318, 4.06281629421142232561092527243, 4.93213569985568165199959255972, 5.79367924138190114320782861346, 7.11962274433598327916875275569, 7.83709052622580173282315018755, 8.957629836994008346834429375009, 9.748754648552596153722677321849

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