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Properties

Label 2-483-7.2-c1-0-27
Degree $2$
Conductor $483$
Sign $-0.991 - 0.126i$
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  + (1.28 − 2.21i)2-s + (−0.5 − 0.866i)3-s + (−2.28 − 3.95i)4-s + (1.28 − 2.21i)5-s − 2.56·6-s + (2.5 − 0.866i)7-s − 6.56·8-s + (−0.499 + 0.866i)9-s + (−3.28 − 5.68i)10-s + (1 + 1.73i)11-s + (−2.28 + 3.95i)12-s + 2.12·13-s + (1.28 − 6.65i)14-s − 2.56·15-s + (−3.84 + 6.65i)16-s + (1.71 + 2.97i)17-s + ⋯
L(s)  = 1  + (0.905 − 1.56i)2-s + (−0.288 − 0.499i)3-s + (−1.14 − 1.97i)4-s + (0.572 − 0.992i)5-s − 1.04·6-s + (0.944 − 0.327i)7-s − 2.31·8-s + (−0.166 + 0.288i)9-s + (−1.03 − 1.79i)10-s + (0.301 + 0.522i)11-s + (−0.658 + 1.14i)12-s + 0.588·13-s + (0.342 − 1.77i)14-s − 0.661·15-s + (−0.960 + 1.66i)16-s + (0.416 + 0.722i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.991 - 0.126i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.144812 + 2.28194i\)
\(L(\frac12)\) \(\approx\) \(0.144812 + 2.28194i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-2.5 + 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-1.28 + 2.21i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.28 + 2.21i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.12T + 13T^{2} \)
17 \( 1 + (-1.71 - 2.97i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.78 - 4.81i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 + 0.876T + 29T^{2} \)
31 \( 1 + (-3.34 - 5.78i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.78 - 6.54i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.24T + 41T^{2} \)
43 \( 1 + 12.6T + 43T^{2} \)
47 \( 1 + (-2.84 + 4.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.40 + 11.0i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.438 + 0.759i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.93 - 5.08i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.80T + 71T^{2} \)
73 \( 1 + (7.06 + 12.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.78 + 8.28i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.1T + 83T^{2} \)
89 \( 1 + (-5 + 8.66i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.36T + 97T^{2} \)
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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

−10.60240803574409269414830699874, −10.13592590602564997231530472004, −8.928654162968726193263440888094, −8.059227927735918481623161656023, −6.36541954870547248708575720442, −5.34091302821197766296043342192, −4.67512297750776724756958899021, −3.60664737231626261627441002031, −1.78568754583796365940005761725, −1.36782226818114186719182331256, 2.80406520882602336697866095711, 4.09460737494508445701393598870, 5.04431152127747262566345262315, 5.93529181365315807535493468796, 6.54605009165360557606150031686, 7.54572658791151997267091225868, 8.517148262296334485792977703246, 9.373203386470896031017301014028, 10.75637206633647383231170114485, 11.40761081887226630971422848659

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