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Properties

Label 2-232-29.9-c1-0-5
Degree $2$
Conductor $232$
Sign $0.718 + 0.695i$
Analytic cond. $1.85252$
Root an. cond. $1.36107$
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  + (0.0130 − 0.0104i)3-s + (3.41 − 1.64i)5-s + (−3.15 − 3.95i)7-s + (−0.667 + 2.92i)9-s + (2.67 − 0.610i)11-s + (−0.666 − 2.92i)13-s + (0.0275 − 0.0572i)15-s + 2.98i·17-s + (2.25 + 1.79i)19-s + (−0.0826 − 0.0188i)21-s + (0.244 + 0.117i)23-s + (5.85 − 7.33i)25-s + (0.0436 + 0.0905i)27-s + (3.83 + 3.77i)29-s + (2.01 + 4.18i)31-s + ⋯
L(s)  = 1  + (0.00756 − 0.00603i)3-s + (1.52 − 0.735i)5-s + (−1.19 − 1.49i)7-s + (−0.222 + 0.974i)9-s + (0.806 − 0.184i)11-s + (−0.184 − 0.810i)13-s + (0.00711 − 0.0147i)15-s + 0.723i·17-s + (0.517 + 0.412i)19-s + (−0.0180 − 0.00411i)21-s + (0.0510 + 0.0245i)23-s + (1.17 − 1.46i)25-s + (0.00839 + 0.0174i)27-s + (0.712 + 0.701i)29-s + (0.362 + 0.752i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(232\)    =    \(2^{3} \cdot 29\)
Sign: $0.718 + 0.695i$
Analytic conductor: \(1.85252\)
Root analytic conductor: \(1.36107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{232} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 232,\ (\ :1/2),\ 0.718 + 0.695i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29422 - 0.523798i\)
\(L(\frac12)\) \(\approx\) \(1.29422 - 0.523798i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 + (-3.83 - 3.77i)T \)
good3 \( 1 + (-0.0130 + 0.0104i)T + (0.667 - 2.92i)T^{2} \)
5 \( 1 + (-3.41 + 1.64i)T + (3.11 - 3.90i)T^{2} \)
7 \( 1 + (3.15 + 3.95i)T + (-1.55 + 6.82i)T^{2} \)
11 \( 1 + (-2.67 + 0.610i)T + (9.91 - 4.77i)T^{2} \)
13 \( 1 + (0.666 + 2.92i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 - 2.98iT - 17T^{2} \)
19 \( 1 + (-2.25 - 1.79i)T + (4.22 + 18.5i)T^{2} \)
23 \( 1 + (-0.244 - 0.117i)T + (14.3 + 17.9i)T^{2} \)
31 \( 1 + (-2.01 - 4.18i)T + (-19.3 + 24.2i)T^{2} \)
37 \( 1 + (5.69 + 1.30i)T + (33.3 + 16.0i)T^{2} \)
41 \( 1 + 4.84iT - 41T^{2} \)
43 \( 1 + (5.11 - 10.6i)T + (-26.8 - 33.6i)T^{2} \)
47 \( 1 + (5.57 - 1.27i)T + (42.3 - 20.3i)T^{2} \)
53 \( 1 + (3.74 - 1.80i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 - 0.250T + 59T^{2} \)
61 \( 1 + (0.554 - 0.441i)T + (13.5 - 59.4i)T^{2} \)
67 \( 1 + (2.02 - 8.88i)T + (-60.3 - 29.0i)T^{2} \)
71 \( 1 + (-0.00380 - 0.0166i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-6.42 + 13.3i)T + (-45.5 - 57.0i)T^{2} \)
79 \( 1 + (-10.6 - 2.42i)T + (71.1 + 34.2i)T^{2} \)
83 \( 1 + (0.963 - 1.20i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (1.16 + 2.41i)T + (-55.4 + 69.5i)T^{2} \)
97 \( 1 + (-5.83 - 4.65i)T + (21.5 + 94.5i)T^{2} \)
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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

−12.43218749447409120924639419206, −10.68464890296078897763687170529, −10.17258865004710621511773664558, −9.381812354081227130565135307715, −8.199235531578415027579536271005, −6.85835019029960845898414516183, −5.94567771318016558203438610991, −4.80352834494604396112921317676, −3.25490201629753992760024821380, −1.36848339653152413346138629236, 2.24526662684880031683423601071, 3.26222685924051742785895871005, 5.34677076714422171139900674542, 6.53450156190817222234562923712, 6.55860591304982385874716716611, 8.903158783502238322512685294804, 9.484360216550798009374273082459, 9.934956388249189730272351617620, 11.61190364179537459536002357678, 12.17050140679681453726779402401

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