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Properties

Label 2-1232-77.54-c1-0-45
Degree $2$
Conductor $1232$
Sign $-0.960 - 0.277i$
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  + (1.58 − 0.912i)3-s + (−3.66 − 2.11i)5-s + (−0.791 − 2.52i)7-s + (0.166 − 0.288i)9-s + (1.46 − 2.97i)11-s + 1.85·13-s − 7.72·15-s + (−2.15 − 3.73i)17-s + (−3.02 + 5.23i)19-s + (−3.55 − 3.26i)21-s + (−3.08 + 5.34i)23-s + (6.45 + 11.1i)25-s + 4.86i·27-s − 6.96i·29-s + (−1.03 + 0.599i)31-s + ⋯
L(s)  = 1  + (0.912 − 0.527i)3-s + (−1.63 − 0.946i)5-s + (−0.299 − 0.954i)7-s + (0.0555 − 0.0961i)9-s + (0.442 − 0.896i)11-s + 0.513·13-s − 1.99·15-s + (−0.523 − 0.906i)17-s + (−0.693 + 1.20i)19-s + (−0.776 − 0.713i)21-s + (−0.643 + 1.11i)23-s + (1.29 + 2.23i)25-s + 0.937i·27-s − 1.29i·29-s + (−0.186 + 0.107i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 - 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $-0.960 - 0.277i$
Analytic conductor: \(9.83756\)
Root analytic conductor: \(3.13649\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1232} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :1/2),\ -0.960 - 0.277i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7539761958\)
\(L(\frac12)\) \(\approx\) \(0.7539761958\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.791 + 2.52i)T \)
11 \( 1 + (-1.46 + 2.97i)T \)
good3 \( 1 + (-1.58 + 0.912i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (3.66 + 2.11i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 1.85T + 13T^{2} \)
17 \( 1 + (2.15 + 3.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.02 - 5.23i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.08 - 5.34i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.96iT - 29T^{2} \)
31 \( 1 + (1.03 - 0.599i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.487 - 0.844i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.41T + 41T^{2} \)
43 \( 1 + 0.828iT - 43T^{2} \)
47 \( 1 + (10.3 + 6.00i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.113 + 0.196i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.558 + 0.322i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.169 + 0.293i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.172 + 0.297i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.04T + 71T^{2} \)
73 \( 1 + (5.52 + 9.57i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.81 + 4.51i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.47T + 83T^{2} \)
89 \( 1 + (-2.17 - 1.25i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 15.9iT - 97T^{2} \)
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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

−8.899957926527764279664472042286, −8.360478505518645487267339940163, −7.77656660115853127654440058170, −7.19699111628943798463570632233, −6.02271009915047692526738884532, −4.67586172846459929757819062882, −3.79116408300507483997211057292, −3.32556075540814645037673247248, −1.56493962075070736115612034990, −0.28401827291759908190243029372, 2.33787970218310074640419520049, 3.15422359372131505279450093914, 3.97578536261450179822145064630, 4.61502304184055951297048400147, 6.34698738674147530903954501574, 6.82208281882404324322517458966, 7.934022172936337973835731645098, 8.605665420733540459613043720551, 9.071958407294788485575571228581, 10.16564162846532076890515608501

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