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Properties

Label 2-483-161.34-c1-0-3
Degree $2$
Conductor $483$
Sign $-0.835 - 0.549i$
Analytic cond. $3.85677$
Root an. cond. $1.96386$
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  + (−0.795 + 0.510i)2-s + (−0.989 − 0.142i)3-s + (−0.459 + 1.00i)4-s + (1.49 − 0.438i)5-s + (0.859 − 0.392i)6-s + (−0.275 + 2.63i)7-s + (−0.417 − 2.90i)8-s + (0.959 + 0.281i)9-s + (−0.964 + 1.11i)10-s + (−0.572 + 0.890i)11-s + (0.598 − 0.931i)12-s + (0.623 + 0.540i)13-s + (−1.12 − 2.23i)14-s + (−1.54 + 0.221i)15-s + (0.367 + 0.424i)16-s + (2.42 + 5.30i)17-s + ⋯
L(s)  = 1  + (−0.562 + 0.361i)2-s + (−0.571 − 0.0821i)3-s + (−0.229 + 0.503i)4-s + (0.668 − 0.196i)5-s + (0.350 − 0.160i)6-s + (−0.104 + 0.994i)7-s + (−0.147 − 1.02i)8-s + (0.319 + 0.0939i)9-s + (−0.304 + 0.351i)10-s + (−0.172 + 0.268i)11-s + (0.172 − 0.268i)12-s + (0.173 + 0.149i)13-s + (−0.300 − 0.596i)14-s + (−0.398 + 0.0572i)15-s + (0.0919 + 0.106i)16-s + (0.587 + 1.28i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.835 - 0.549i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.835 - 0.549i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.179865 + 0.600760i\)
\(L(\frac12)\) \(\approx\) \(0.179865 + 0.600760i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.989 + 0.142i)T \)
7 \( 1 + (0.275 - 2.63i)T \)
23 \( 1 + (3.24 + 3.53i)T \)
good2 \( 1 + (0.795 - 0.510i)T + (0.830 - 1.81i)T^{2} \)
5 \( 1 + (-1.49 + 0.438i)T + (4.20 - 2.70i)T^{2} \)
11 \( 1 + (0.572 - 0.890i)T + (-4.56 - 10.0i)T^{2} \)
13 \( 1 + (-0.623 - 0.540i)T + (1.85 + 12.8i)T^{2} \)
17 \( 1 + (-2.42 - 5.30i)T + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (0.166 - 0.365i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (0.929 + 2.03i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (6.11 - 0.879i)T + (29.7 - 8.73i)T^{2} \)
37 \( 1 + (3.05 - 10.4i)T + (-31.1 - 20.0i)T^{2} \)
41 \( 1 + (-3.48 - 11.8i)T + (-34.4 + 22.1i)T^{2} \)
43 \( 1 + (11.9 + 1.72i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 + 4.76iT - 47T^{2} \)
53 \( 1 + (4.79 - 4.15i)T + (7.54 - 52.4i)T^{2} \)
59 \( 1 + (-4.32 - 3.75i)T + (8.39 + 58.3i)T^{2} \)
61 \( 1 + (1.11 + 7.77i)T + (-58.5 + 17.1i)T^{2} \)
67 \( 1 + (-0.784 - 1.22i)T + (-27.8 + 60.9i)T^{2} \)
71 \( 1 + (-8.54 + 5.49i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (-13.8 - 6.31i)T + (47.8 + 55.1i)T^{2} \)
79 \( 1 + (-1.45 - 1.26i)T + (11.2 + 78.1i)T^{2} \)
83 \( 1 + (-4.40 - 1.29i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (-1.25 + 8.74i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (-4.81 + 1.41i)T + (81.6 - 52.4i)T^{2} \)
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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

−11.41110769864461004001793628258, −10.09683546570100924451081389593, −9.625406750153838166565723080849, −8.533816671005681155403307464471, −7.967358498881320976679029434580, −6.61755666624119019218581966435, −5.97087856653833882906215539966, −4.87799572540118257999003261856, −3.48895885280388576681538601869, −1.81089809163576564219354067513, 0.49232777349782050329963510547, 1.95158315545756754142529190087, 3.67553064674721680023749848349, 5.13665248637570571040410984541, 5.76680954191116078054885633038, 6.94305265390703369427461848745, 7.914581602982989965674570324509, 9.293309324333645270412369991200, 9.773302124135505568515198087619, 10.62695650628894896095650492840

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