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Properties

Label 2-232-232.115-c2-0-11
Degree $2$
Conductor $232$
Sign $0.386 - 0.922i$
Analytic cond. $6.32154$
Root an. cond. $2.51426$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.64 − 1.14i)2-s + 1.23i·3-s + (1.39 + 3.74i)4-s + 6.52i·5-s + (1.41 − 2.03i)6-s − 8.84i·7-s + (1.98 − 7.75i)8-s + 7.47·9-s + (7.44 − 10.7i)10-s − 8.03i·11-s + (−4.63 + 1.72i)12-s + 23.9i·13-s + (−10.0 + 14.5i)14-s − 8.07·15-s + (−12.1 + 10.4i)16-s + 3.61i·17-s + ⋯
L(s)  = 1  + (−0.821 − 0.570i)2-s + 0.412i·3-s + (0.349 + 0.937i)4-s + 1.30i·5-s + (0.235 − 0.338i)6-s − 1.26i·7-s + (0.247 − 0.968i)8-s + 0.830·9-s + (0.744 − 1.07i)10-s − 0.730i·11-s + (−0.386 + 0.143i)12-s + 1.84i·13-s + (−0.720 + 1.03i)14-s − 0.538·15-s + (−0.756 + 0.654i)16-s + 0.212i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(232\)    =    \(2^{3} \cdot 29\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(6.32154\)
Root analytic conductor: \(2.51426\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{232} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 232,\ (\ :1),\ 0.386 - 0.922i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.829572 + 0.551904i\)
\(L(\frac12)\) \(\approx\) \(0.829572 + 0.551904i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.64 + 1.14i)T \)
29 \( 1 + (28.6 + 4.22i)T \)
good3 \( 1 - 1.23iT - 9T^{2} \)
5 \( 1 - 6.52iT - 25T^{2} \)
7 \( 1 + 8.84iT - 49T^{2} \)
11 \( 1 + 8.03iT - 121T^{2} \)
13 \( 1 - 23.9iT - 169T^{2} \)
17 \( 1 - 3.61iT - 289T^{2} \)
19 \( 1 - 25.4iT - 361T^{2} \)
23 \( 1 - 12.7iT - 529T^{2} \)
31 \( 1 - 31.7T + 961T^{2} \)
37 \( 1 + 28.0T + 1.36e3T^{2} \)
41 \( 1 - 58.3iT - 1.68e3T^{2} \)
43 \( 1 - 20.5iT - 1.84e3T^{2} \)
47 \( 1 + 53.0T + 2.20e3T^{2} \)
53 \( 1 + 77.4iT - 2.80e3T^{2} \)
59 \( 1 - 36.8T + 3.48e3T^{2} \)
61 \( 1 - 114.T + 3.72e3T^{2} \)
67 \( 1 - 21.5T + 4.48e3T^{2} \)
71 \( 1 + 18.8iT - 5.04e3T^{2} \)
73 \( 1 - 105. iT - 5.32e3T^{2} \)
79 \( 1 - 17.8T + 6.24e3T^{2} \)
83 \( 1 + 0.565T + 6.88e3T^{2} \)
89 \( 1 + 91.6iT - 7.92e3T^{2} \)
97 \( 1 + 86.0iT - 9.40e3T^{2} \)
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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

−11.56812262720111844251619756098, −11.10074006015837183120686739615, −10.08583838056218060634666355360, −9.745552589704863982439337406815, −8.220395882090359782121396452040, −7.12743349829159376916233198213, −6.57755426059994605214829098093, −4.14484594421968188898083407961, −3.46738305925007859156525994340, −1.64498168577885890919552892830, 0.74424983403998985637402751118, 2.27076854780717177685150318920, 4.88565642132088205466898972094, 5.55253638198087865967904800077, 6.91771903663895342136219925989, 7.980512596257577718642276314385, 8.762804267149746949735607472876, 9.538988737733042310287720390218, 10.55339319461833791888972090392, 11.99559174250818139435330959308

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