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Properties

Label 2-1232-7.2-c1-0-32
Degree $2$
Conductor $1232$
Sign $0.872 + 0.489i$
Analytic cond. $9.83756$
Root an. cond. $3.13649$
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  + (0.703 + 1.21i)3-s + (1.30 − 2.26i)5-s + (2.64 + 0.135i)7-s + (0.508 − 0.881i)9-s + (−0.5 − 0.866i)11-s + 1.40·13-s + 3.67·15-s + (−3.54 − 6.14i)17-s + (−0.0616 + 0.106i)19-s + (1.69 + 3.31i)21-s + (1.10 − 1.90i)23-s + (−0.906 − 1.56i)25-s + 5.65·27-s − 4.42·29-s + (−0.734 − 1.27i)31-s + ⋯
L(s)  = 1  + (0.406 + 0.703i)3-s + (0.583 − 1.01i)5-s + (0.998 + 0.0511i)7-s + (0.169 − 0.293i)9-s + (−0.150 − 0.261i)11-s + 0.390·13-s + 0.948·15-s + (−0.860 − 1.49i)17-s + (−0.0141 + 0.0244i)19-s + (0.369 + 0.723i)21-s + (0.229 − 0.397i)23-s + (−0.181 − 0.313i)25-s + 1.08·27-s − 0.821·29-s + (−0.131 − 0.228i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $0.872 + 0.489i$
Analytic conductor: \(9.83756\)
Root analytic conductor: \(3.13649\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1232} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :1/2),\ 0.872 + 0.489i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.341165543\)
\(L(\frac12)\) \(\approx\) \(2.341165543\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-2.64 - 0.135i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (-0.703 - 1.21i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.30 + 2.26i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 1.40T + 13T^{2} \)
17 \( 1 + (3.54 + 6.14i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.0616 - 0.106i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.10 + 1.90i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.42T + 29T^{2} \)
31 \( 1 + (0.734 + 1.27i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.610 - 1.05i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.70T + 41T^{2} \)
43 \( 1 - 1.59T + 43T^{2} \)
47 \( 1 + (5.14 - 8.90i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.633 + 1.09i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.151 - 0.261i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.13 + 5.43i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.14 - 10.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.09T + 71T^{2} \)
73 \( 1 + (-2.47 - 4.27i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.99 + 5.18i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.31T + 83T^{2} \)
89 \( 1 + (-5.36 + 9.28i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.1T + 97T^{2} \)
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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

−9.378458467045006273846306026976, −9.005213457612362958959123623779, −8.322381928110431245011371843777, −7.29609201013839106384176933757, −6.18725816278852714352959866991, −5.02124662316650491339122782781, −4.74336505673123469483537559887, −3.59955459734078911678920968015, −2.29819004951425381747512745795, −1.02383531020239759713313240007, 1.70720351239590826507889529511, 2.16632103221284651471234788246, 3.48546472257027595229508566709, 4.65310856730098288132762462709, 5.72364610943472233659891060282, 6.65966158102791521452445738809, 7.26800248133409040948641005665, 8.145943469866788532212197323159, 8.709639743102100927033559354739, 9.913261369095019983026241695516

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