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Properties

Label 2-1232-7.2-c1-0-15
Degree $2$
Conductor $1232$
Sign $0.132 + 0.991i$
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  + (−1.66 − 2.87i)3-s + (−0.852 + 1.47i)5-s + (1.47 + 2.19i)7-s + (−4.01 + 6.95i)9-s + (−0.5 − 0.866i)11-s − 3.32·13-s + 5.66·15-s + (−0.433 − 0.751i)17-s + (1.13 − 1.97i)19-s + (3.85 − 7.89i)21-s + (1.30 − 2.26i)23-s + (1.04 + 1.81i)25-s + 16.6·27-s + 9.34·29-s + (−4.29 − 7.44i)31-s + ⋯
L(s)  = 1  + (−0.958 − 1.66i)3-s + (−0.381 + 0.660i)5-s + (0.559 + 0.829i)7-s + (−1.33 + 2.31i)9-s + (−0.150 − 0.261i)11-s − 0.920·13-s + 1.46·15-s + (−0.105 − 0.182i)17-s + (0.261 − 0.452i)19-s + (0.840 − 1.72i)21-s + (0.272 − 0.472i)23-s + (0.209 + 0.362i)25-s + 3.21·27-s + 1.73·29-s + (−0.772 − 1.33i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.132 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.132 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $0.132 + 0.991i$
Analytic conductor: \(9.83756\)
Root analytic conductor: \(3.13649\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1232} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :1/2),\ 0.132 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9274121146\)
\(L(\frac12)\) \(\approx\) \(0.9274121146\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.47 - 2.19i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (1.66 + 2.87i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.852 - 1.47i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 3.32T + 13T^{2} \)
17 \( 1 + (0.433 + 0.751i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.13 + 1.97i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.30 + 2.26i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 9.34T + 29T^{2} \)
31 \( 1 + (4.29 + 7.44i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.70 + 6.41i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.83T + 41T^{2} \)
43 \( 1 - 6.32T + 43T^{2} \)
47 \( 1 + (1.20 - 2.08i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.99 + 6.91i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.53 + 9.58i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.71 + 6.43i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.24 - 5.61i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.867T + 71T^{2} \)
73 \( 1 + (-6.82 - 11.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.73 + 8.20i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.54T + 83T^{2} \)
89 \( 1 + (4.43 - 7.67i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.2T + 97T^{2} \)
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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

−9.476819346940960919393839427054, −8.346732692722136338993116518062, −7.69198111872119640868903788028, −7.07401356608243127544291362718, −6.29948451919958539777024859259, −5.50515250047377203577266254742, −4.71829541858071856749549333768, −2.78990708129728574701902862356, −2.13912481872094602978614318627, −0.61374812584920777486864902610, 0.913792783683382406522894114100, 3.11870686691586069998843844887, 4.26366044517993092306526240656, 4.66399273093675541132387875094, 5.30613732097528983823645369295, 6.39087020173281017751153344430, 7.47425319051984043619672926116, 8.479181707162433873314913931242, 9.281026614286947494571866926739, 10.10942250497809528454475491575

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