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Properties

Label 2-232-232.109-c1-0-5
Degree $2$
Conductor $232$
Sign $0.0313 - 0.999i$
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  + (0.763 + 1.19i)2-s + (−2.67 − 1.28i)3-s + (−0.835 + 1.81i)4-s + (1.61 − 0.369i)5-s + (−0.506 − 4.16i)6-s + (2.81 + 1.35i)7-s + (−2.80 + 0.391i)8-s + (3.61 + 4.53i)9-s + (1.67 + 1.64i)10-s + (−3.79 + 4.75i)11-s + (4.57 − 3.78i)12-s + (3.45 + 2.75i)13-s + (0.533 + 4.38i)14-s + (−4.80 − 1.09i)15-s + (−2.60 − 3.03i)16-s + 0.692i·17-s + ⋯
L(s)  = 1  + (0.539 + 0.841i)2-s + (−1.54 − 0.743i)3-s + (−0.417 + 0.908i)4-s + (0.724 − 0.165i)5-s + (−0.206 − 1.70i)6-s + (1.06 + 0.512i)7-s + (−0.990 + 0.138i)8-s + (1.20 + 1.51i)9-s + (0.529 + 0.520i)10-s + (−1.14 + 1.43i)11-s + (1.32 − 1.09i)12-s + (0.957 + 0.763i)13-s + (0.142 + 1.17i)14-s + (−1.24 − 0.283i)15-s + (−0.650 − 0.759i)16-s + 0.167i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0313 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0313 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(232\)    =    \(2^{3} \cdot 29\)
Sign: $0.0313 - 0.999i$
Analytic conductor: \(1.85252\)
Root analytic conductor: \(1.36107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{232} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 232,\ (\ :1/2),\ 0.0313 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.791959 + 0.767480i\)
\(L(\frac12)\) \(\approx\) \(0.791959 + 0.767480i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.763 - 1.19i)T \)
29 \( 1 + (-4.46 + 3.01i)T \)
good3 \( 1 + (2.67 + 1.28i)T + (1.87 + 2.34i)T^{2} \)
5 \( 1 + (-1.61 + 0.369i)T + (4.50 - 2.16i)T^{2} \)
7 \( 1 + (-2.81 - 1.35i)T + (4.36 + 5.47i)T^{2} \)
11 \( 1 + (3.79 - 4.75i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (-3.45 - 2.75i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 - 0.692iT - 17T^{2} \)
19 \( 1 + (-4.80 + 2.31i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (-0.102 + 0.449i)T + (-20.7 - 9.97i)T^{2} \)
31 \( 1 + (3.19 - 0.728i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 + (-3.16 - 3.96i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + 6.49iT - 41T^{2} \)
43 \( 1 + (-1.17 + 5.14i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (1.04 + 0.829i)T + (10.4 + 45.8i)T^{2} \)
53 \( 1 + (10.8 - 2.47i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 - 3.70iT - 59T^{2} \)
61 \( 1 + (9.06 + 4.36i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 + (-9.18 + 7.32i)T + (14.9 - 65.3i)T^{2} \)
71 \( 1 + (-1.83 + 2.30i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-1.10 - 0.251i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 + (-3.25 + 2.59i)T + (17.5 - 77.0i)T^{2} \)
83 \( 1 + (-1.99 - 4.13i)T + (-51.7 + 64.8i)T^{2} \)
89 \( 1 + (-0.587 + 0.134i)T + (80.1 - 38.6i)T^{2} \)
97 \( 1 + (-5.41 - 11.2i)T + (-60.4 + 75.8i)T^{2} \)
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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

−12.41973168813015562292927267714, −11.77998990167207412597843175490, −10.84522611321858694729110311176, −9.461933065953461712253144328304, −8.030633342739395748144423684505, −7.16859424624902564322389518172, −6.15900159364781961276167112239, −5.27738897055568572393373746813, −4.71610704096606004605855257640, −1.92854850302052750613214383110, 1.02383795530898876718951541547, 3.32214366361719847062510173835, 4.74967909810487658924394761567, 5.57833165531576130112671057410, 6.09022495276318107444293542220, 8.096088446647437727314506564424, 9.656961949393150614554669387152, 10.52668992839582360595753881452, 11.01058897750361359727964193539, 11.52249517107664817380218410671

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