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Properties

Label 2-232-232.109-c1-0-12
Degree $2$
Conductor $232$
Sign $-0.202 - 0.979i$
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  + (0.461 + 1.33i)2-s + (2.40 + 1.15i)3-s + (−1.57 + 1.23i)4-s + (0.540 − 0.123i)5-s + (−0.439 + 3.75i)6-s + (−1.17 − 0.565i)7-s + (−2.37 − 1.53i)8-s + (2.58 + 3.23i)9-s + (0.414 + 0.665i)10-s + (0.760 − 0.953i)11-s + (−5.22 + 1.14i)12-s + (−1.60 − 1.27i)13-s + (0.214 − 1.82i)14-s + (1.44 + 0.329i)15-s + (0.956 − 3.88i)16-s + 0.926i·17-s + ⋯
L(s)  = 1  + (0.326 + 0.945i)2-s + (1.39 + 0.669i)3-s + (−0.787 + 0.616i)4-s + (0.241 − 0.0551i)5-s + (−0.179 + 1.53i)6-s + (−0.443 − 0.213i)7-s + (−0.839 − 0.542i)8-s + (0.861 + 1.07i)9-s + (0.130 + 0.210i)10-s + (0.229 − 0.287i)11-s + (−1.50 + 0.330i)12-s + (−0.445 − 0.354i)13-s + (0.0571 − 0.488i)14-s + (0.372 + 0.0851i)15-s + (0.239 − 0.970i)16-s + 0.224i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.202 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.202 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(232\)    =    \(2^{3} \cdot 29\)
Sign: $-0.202 - 0.979i$
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: \((2,\ 232,\ (\ :1/2),\ -0.202 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22795 + 1.50810i\)
\(L(\frac12)\) \(\approx\) \(1.22795 + 1.50810i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.461 - 1.33i)T \)
29 \( 1 + (-0.151 + 5.38i)T \)
good3 \( 1 + (-2.40 - 1.15i)T + (1.87 + 2.34i)T^{2} \)
5 \( 1 + (-0.540 + 0.123i)T + (4.50 - 2.16i)T^{2} \)
7 \( 1 + (1.17 + 0.565i)T + (4.36 + 5.47i)T^{2} \)
11 \( 1 + (-0.760 + 0.953i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (1.60 + 1.27i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 - 0.926iT - 17T^{2} \)
19 \( 1 + (-4.27 + 2.06i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (-0.344 + 1.50i)T + (-20.7 - 9.97i)T^{2} \)
31 \( 1 + (4.24 - 0.969i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 + (-6.15 - 7.72i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 - 10.8iT - 41T^{2} \)
43 \( 1 + (-0.392 + 1.72i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (6.26 + 4.99i)T + (10.4 + 45.8i)T^{2} \)
53 \( 1 + (4.55 - 1.03i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + 8.00iT - 59T^{2} \)
61 \( 1 + (-10.7 - 5.19i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 + (2.94 - 2.34i)T + (14.9 - 65.3i)T^{2} \)
71 \( 1 + (0.766 - 0.961i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-10.2 - 2.33i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 + (-6.14 + 4.90i)T + (17.5 - 77.0i)T^{2} \)
83 \( 1 + (4.73 + 9.83i)T + (-51.7 + 64.8i)T^{2} \)
89 \( 1 + (13.4 - 3.07i)T + (80.1 - 38.6i)T^{2} \)
97 \( 1 + (4.78 + 9.94i)T + (-60.4 + 75.8i)T^{2} \)
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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

−13.10274504610651085622591046120, −11.62495537871773619900576575966, −9.831318524944007156676326626366, −9.563422820710520325061882787254, −8.404176913297568820184566108128, −7.69750929238797414279222283721, −6.44935968256679648735429120828, −5.07769093167505040435517865232, −3.86630028178647602568832868749, −2.91572660654291044670380816715, 1.76230543186214515470105047382, 2.84613110399685176204284474810, 3.91991100920303172373939289160, 5.56860893187637477601559889153, 7.03642846848423314433848520860, 8.136475421831919661115821716047, 9.435748007225586308462343690061, 9.532554005135134840722827107784, 11.06451357542868907731983996461, 12.28527880107886064664858670831

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