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Properties

Label 2-232-29.22-c1-0-4
Degree $2$
Conductor $232$
Sign $0.888 + 0.458i$
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  + (0.201 − 0.417i)3-s + (−0.296 − 1.29i)5-s + (2.78 + 1.34i)7-s + (1.73 + 2.17i)9-s + (−2.85 − 2.27i)11-s + (4.30 − 5.40i)13-s + (−0.602 − 0.137i)15-s − 3.49i·17-s + (0.431 + 0.895i)19-s + (1.12 − 0.893i)21-s + (−2.01 + 8.84i)23-s + (2.90 − 1.39i)25-s + (2.61 − 0.597i)27-s + (−2.02 + 4.99i)29-s + (−7.11 + 1.62i)31-s + ⋯
L(s)  = 1  + (0.116 − 0.241i)3-s + (−0.132 − 0.581i)5-s + (1.05 + 0.506i)7-s + (0.578 + 0.725i)9-s + (−0.859 − 0.685i)11-s + (1.19 − 1.49i)13-s + (−0.155 − 0.0355i)15-s − 0.848i·17-s + (0.0989 + 0.205i)19-s + (0.244 − 0.194i)21-s + (−0.421 + 1.84i)23-s + (0.580 − 0.279i)25-s + (0.503 − 0.114i)27-s + (−0.375 + 0.926i)29-s + (−1.27 + 0.291i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(232\)    =    \(2^{3} \cdot 29\)
Sign: $0.888 + 0.458i$
Analytic conductor: \(1.85252\)
Root analytic conductor: \(1.36107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{232} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 232,\ (\ :1/2),\ 0.888 + 0.458i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34919 - 0.327443i\)
\(L(\frac12)\) \(\approx\) \(1.34919 - 0.327443i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.02 - 4.99i)T \)
good3 \( 1 + (-0.201 + 0.417i)T + (-1.87 - 2.34i)T^{2} \)
5 \( 1 + (0.296 + 1.29i)T + (-4.50 + 2.16i)T^{2} \)
7 \( 1 + (-2.78 - 1.34i)T + (4.36 + 5.47i)T^{2} \)
11 \( 1 + (2.85 + 2.27i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-4.30 + 5.40i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + 3.49iT - 17T^{2} \)
19 \( 1 + (-0.431 - 0.895i)T + (-11.8 + 14.8i)T^{2} \)
23 \( 1 + (2.01 - 8.84i)T + (-20.7 - 9.97i)T^{2} \)
31 \( 1 + (7.11 - 1.62i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 + (6.57 - 5.23i)T + (8.23 - 36.0i)T^{2} \)
41 \( 1 + 1.91iT - 41T^{2} \)
43 \( 1 + (-1.99 - 0.455i)T + (38.7 + 18.6i)T^{2} \)
47 \( 1 + (4.87 + 3.88i)T + (10.4 + 45.8i)T^{2} \)
53 \( 1 + (-0.910 - 3.99i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + 4.78T + 59T^{2} \)
61 \( 1 + (2.33 - 4.85i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 + (-1.90 - 2.38i)T + (-14.9 + 65.3i)T^{2} \)
71 \( 1 + (3.50 - 4.39i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (10.6 + 2.44i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 + (-11.7 + 9.39i)T + (17.5 - 77.0i)T^{2} \)
83 \( 1 + (-2.76 + 1.33i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (-10.1 + 2.30i)T + (80.1 - 38.6i)T^{2} \)
97 \( 1 + (-4.58 - 9.51i)T + (-60.4 + 75.8i)T^{2} \)
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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

−12.16056452133082998523254620902, −11.08901550431319816168244924697, −10.43136310263248407062561761499, −8.904640859471766765288559992089, −8.135316086663529801235165753823, −7.46938464600313241357240825395, −5.55561296154204136408949903159, −5.07848695978735214631466654522, −3.27975610194102889668633240323, −1.49976189745920633126969286224, 1.87102812620924811894296985823, 3.81775111362203081176315665056, 4.62662606553982119019970485412, 6.32711212307476114679421750473, 7.22770200315879432099131766776, 8.318112140119484388059359299859, 9.361504530218936596243237614704, 10.61138095208965595515178518412, 11.01961691292307578902180396178, 12.22934377247273445694670766873

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