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Properties

Label 2-1232-308.263-c1-0-42
Degree $2$
Conductor $1232$
Sign $-0.978 + 0.207i$
Analytic cond. $9.83756$
Root an. cond. $3.13649$
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  + (−0.497 − 0.287i)3-s + (0.0931 + 0.161i)5-s + (1.17 − 2.36i)7-s + (−1.33 − 2.31i)9-s + (−2.46 + 2.22i)11-s − 2.60i·13-s − 0.106i·15-s + (−2.71 − 1.56i)17-s + (1.90 + 3.29i)19-s + (−1.26 + 0.839i)21-s + (−6.10 + 3.52i)23-s + (2.48 − 4.30i)25-s + 3.25i·27-s + 2.51i·29-s + (−7.39 − 4.26i)31-s + ⋯
L(s)  = 1  + (−0.287 − 0.165i)3-s + (0.0416 + 0.0721i)5-s + (0.445 − 0.895i)7-s + (−0.445 − 0.770i)9-s + (−0.742 + 0.669i)11-s − 0.723i·13-s − 0.0276i·15-s + (−0.659 − 0.380i)17-s + (0.436 + 0.756i)19-s + (−0.276 + 0.183i)21-s + (−1.27 + 0.734i)23-s + (0.496 − 0.860i)25-s + 0.626i·27-s + 0.467i·29-s + (−1.32 − 0.766i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $-0.978 + 0.207i$
Analytic conductor: \(9.83756\)
Root analytic conductor: \(3.13649\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1232} (879, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :1/2),\ -0.978 + 0.207i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5065285291\)
\(L(\frac12)\) \(\approx\) \(0.5065285291\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-1.17 + 2.36i)T \)
11 \( 1 + (2.46 - 2.22i)T \)
good3 \( 1 + (0.497 + 0.287i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.0931 - 0.161i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 2.60iT - 13T^{2} \)
17 \( 1 + (2.71 + 1.56i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.90 - 3.29i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.10 - 3.52i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.51iT - 29T^{2} \)
31 \( 1 + (7.39 + 4.26i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.18 + 2.04i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.66iT - 41T^{2} \)
43 \( 1 + 2.06T + 43T^{2} \)
47 \( 1 + (4.65 - 2.68i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.47 + 6.01i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.586 - 0.338i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.14 + 3.54i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.1 + 5.84i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.20iT - 71T^{2} \)
73 \( 1 + (8.45 + 4.88i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.83 - 10.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.452T + 83T^{2} \)
89 \( 1 + (6.54 + 11.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.54T + 97T^{2} \)
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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

−9.540718462672819259610738881796, −8.328621871169952884377249638329, −7.69113592943083031695403462126, −6.92343045743693659210873640362, −5.96122962184131811704858975930, −5.12983136371887720437131546696, −4.11560003150465821373785521168, −3.11520656957124611615170598552, −1.72147339922867936183218640680, −0.20913669981092993813006910524, 1.91814163540713775245633560400, 2.79224270948073357654351579911, 4.19191848977886088850926601634, 5.23636378096562870987686598353, 5.63737949310868030352229971206, 6.73489431878988951909071276998, 7.77084032522998220667985487635, 8.633040065910107325273466957016, 9.016007270265713590111588099231, 10.26096266769497222626446123933

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