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Properties

Label 2-232-29.13-c1-0-7
Degree $2$
Conductor $232$
Sign $-0.930 + 0.366i$
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  + (−0.728 − 0.580i)3-s + (−2.19 − 1.05i)5-s + (−0.988 + 1.23i)7-s + (−0.474 − 2.07i)9-s + (−4.61 − 1.05i)11-s + (−0.306 + 1.34i)13-s + (0.984 + 2.04i)15-s − 5.43i·17-s + (−5.11 + 4.08i)19-s + (1.43 − 0.328i)21-s + (2.65 − 1.27i)23-s + (0.581 + 0.729i)25-s + (−2.07 + 4.30i)27-s + (4.99 − 2.01i)29-s + (1.68 − 3.50i)31-s + ⋯
L(s)  = 1  + (−0.420 − 0.335i)3-s + (−0.981 − 0.472i)5-s + (−0.373 + 0.468i)7-s + (−0.158 − 0.693i)9-s + (−1.39 − 0.317i)11-s + (−0.0851 + 0.372i)13-s + (0.254 + 0.527i)15-s − 1.31i·17-s + (−1.17 + 0.936i)19-s + (0.313 − 0.0716i)21-s + (0.553 − 0.266i)23-s + (0.116 + 0.145i)25-s + (−0.399 + 0.828i)27-s + (0.927 − 0.374i)29-s + (0.302 − 0.629i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.930 + 0.366i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.930 + 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(232\)    =    \(2^{3} \cdot 29\)
Sign: $-0.930 + 0.366i$
Analytic conductor: \(1.85252\)
Root analytic conductor: \(1.36107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{232} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 232,\ (\ :1/2),\ -0.930 + 0.366i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0654870 - 0.344538i\)
\(L(\frac12)\) \(\approx\) \(0.0654870 - 0.344538i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.99 + 2.01i)T \)
good3 \( 1 + (0.728 + 0.580i)T + (0.667 + 2.92i)T^{2} \)
5 \( 1 + (2.19 + 1.05i)T + (3.11 + 3.90i)T^{2} \)
7 \( 1 + (0.988 - 1.23i)T + (-1.55 - 6.82i)T^{2} \)
11 \( 1 + (4.61 + 1.05i)T + (9.91 + 4.77i)T^{2} \)
13 \( 1 + (0.306 - 1.34i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + 5.43iT - 17T^{2} \)
19 \( 1 + (5.11 - 4.08i)T + (4.22 - 18.5i)T^{2} \)
23 \( 1 + (-2.65 + 1.27i)T + (14.3 - 17.9i)T^{2} \)
31 \( 1 + (-1.68 + 3.50i)T + (-19.3 - 24.2i)T^{2} \)
37 \( 1 + (-3.45 + 0.789i)T + (33.3 - 16.0i)T^{2} \)
41 \( 1 - 3.07iT - 41T^{2} \)
43 \( 1 + (2.94 + 6.11i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (-4.72 - 1.07i)T + (42.3 + 20.3i)T^{2} \)
53 \( 1 + (11.5 + 5.57i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 - 7.93T + 59T^{2} \)
61 \( 1 + (-7.18 - 5.72i)T + (13.5 + 59.4i)T^{2} \)
67 \( 1 + (1.15 + 5.07i)T + (-60.3 + 29.0i)T^{2} \)
71 \( 1 + (0.0363 - 0.159i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (3.55 + 7.38i)T + (-45.5 + 57.0i)T^{2} \)
79 \( 1 + (8.09 - 1.84i)T + (71.1 - 34.2i)T^{2} \)
83 \( 1 + (4.07 + 5.11i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (4.51 - 9.38i)T + (-55.4 - 69.5i)T^{2} \)
97 \( 1 + (2.77 - 2.21i)T + (21.5 - 94.5i)T^{2} \)
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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

−11.92713607340816673240106784021, −11.05594719690890903500247925466, −9.819956401818733313056541417516, −8.673967519331349896096568568719, −7.84975011180644413858288649088, −6.66728120866105892808168454547, −5.56976211031207207807910505635, −4.33030482238956459148025804710, −2.81463885214422991803444836573, −0.28628821792644613706435478284, 2.76440939476159578203875739738, 4.18061794551874955796440587406, 5.22606667313156230314256887466, 6.64330673256354372236940414464, 7.70329285409575793508392471367, 8.455261536643848942065596653552, 10.19624616811687937645740238023, 10.65580049710082682749210395761, 11.37975003526302754321011601838, 12.72212405185867711454363043823

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