Properties

Label 2-1232-7.4-c1-0-10
Degree $2$
Conductor $1232$
Sign $-0.991 + 0.126i$
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  + (−1.5 + 2.59i)3-s + (2 + 3.46i)5-s + (2.5 + 0.866i)7-s + (−3 − 5.19i)9-s + (−0.5 + 0.866i)11-s − 13-s − 12·15-s + (−1 + 1.73i)17-s + (3 + 5.19i)19-s + (−6 + 5.19i)21-s + (−1 − 1.73i)23-s + (−5.49 + 9.52i)25-s + 9·27-s + 29-s + (2 − 3.46i)31-s + ⋯
L(s)  = 1  + (−0.866 + 1.49i)3-s + (0.894 + 1.54i)5-s + (0.944 + 0.327i)7-s + (−1 − 1.73i)9-s + (−0.150 + 0.261i)11-s − 0.277·13-s − 3.09·15-s + (−0.242 + 0.420i)17-s + (0.688 + 1.19i)19-s + (−1.30 + 1.13i)21-s + (−0.208 − 0.361i)23-s + (−1.09 + 1.90i)25-s + 1.73·27-s + 0.185·29-s + (0.359 − 0.622i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $-0.991 + 0.126i$
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: \((2,\ 1232,\ (\ :1/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.397971650\)
\(L(\frac12)\) \(\approx\) \(1.397971650\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (1.5 - 2.59i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2 - 3.46i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-1 - 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.5 + 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.5 - 7.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.5 + 12.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5T + 97T^{2} \)
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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

−10.08638981230815411252186910381, −9.882296990117661441077467419163, −8.734944525747083826363749798410, −7.62170568285111875837091377230, −6.52001608261861316084556776059, −5.81673251503624185953993677645, −5.18526133747664355758161892520, −4.17910340465460651190994476003, −3.18201034635913083907509214112, −2.01069426022908631418732977729, 0.70056852305734365499761284713, 1.42310065392705950443276634174, 2.41263048848555210858096599113, 4.59396525793796595078759377340, 5.21753979419448129231833051443, 5.78049644404974675364634348466, 6.84535999236383844040241066768, 7.57434346401167158391202030178, 8.412176934509938844877877507540, 9.067749246381431336561491790921

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