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Properties

Label 2-1232-7.4-c1-0-17
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: $\chi_{1232} (529, \cdot )$
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

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

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