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Properties

Label 2-483-69.68-c1-0-9
Degree $2$
Conductor $483$
Sign $-0.407 - 0.913i$
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  + 2.27i·2-s + (−0.307 − 1.70i)3-s − 3.19·4-s + 0.401·5-s + (3.88 − 0.700i)6-s + i·7-s − 2.71i·8-s + (−2.81 + 1.04i)9-s + 0.914i·10-s + 4.07·11-s + (0.981 + 5.43i)12-s + 1.08·13-s − 2.27·14-s + (−0.123 − 0.684i)15-s − 0.198·16-s + 5.98·17-s + ⋯
L(s)  = 1  + 1.61i·2-s + (−0.177 − 0.984i)3-s − 1.59·4-s + 0.179·5-s + (1.58 − 0.286i)6-s + 0.377i·7-s − 0.959i·8-s + (−0.936 + 0.349i)9-s + 0.289i·10-s + 1.22·11-s + (0.283 + 1.57i)12-s + 0.300·13-s − 0.608·14-s + (−0.0318 − 0.176i)15-s − 0.0495·16-s + 1.45·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.407 - 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.407 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.407 - 0.913i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (344, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.407 - 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.706993 + 1.08968i\)
\(L(\frac12)\) \(\approx\) \(0.706993 + 1.08968i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.307 + 1.70i)T \)
7 \( 1 - iT \)
23 \( 1 + (-1.14 - 4.65i)T \)
good2 \( 1 - 2.27iT - 2T^{2} \)
5 \( 1 - 0.401T + 5T^{2} \)
11 \( 1 - 4.07T + 11T^{2} \)
13 \( 1 - 1.08T + 13T^{2} \)
17 \( 1 - 5.98T + 17T^{2} \)
19 \( 1 - 7.44iT - 19T^{2} \)
29 \( 1 + 4.02iT - 29T^{2} \)
31 \( 1 - 2.95T + 31T^{2} \)
37 \( 1 - 11.2iT - 37T^{2} \)
41 \( 1 - 7.74iT - 41T^{2} \)
43 \( 1 + 6.27iT - 43T^{2} \)
47 \( 1 + 5.04iT - 47T^{2} \)
53 \( 1 - 4.33T + 53T^{2} \)
59 \( 1 - 1.77iT - 59T^{2} \)
61 \( 1 + 6.21iT - 61T^{2} \)
67 \( 1 + 11.4iT - 67T^{2} \)
71 \( 1 + 12.7iT - 71T^{2} \)
73 \( 1 + 1.30T + 73T^{2} \)
79 \( 1 + 11.9iT - 79T^{2} \)
83 \( 1 + 7.67T + 83T^{2} \)
89 \( 1 - 13.4T + 89T^{2} \)
97 \( 1 - 4.86iT - 97T^{2} \)
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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

−11.78550613970130692909659899535, −10.04622337854655637348373210152, −9.114792759460956573799659674709, −8.048993574725782844429956375354, −7.71954439205010351130124395040, −6.44643306717138728571647043775, −6.06856661171601457800038291484, −5.19830835795319132188700310732, −3.59258982917745270659819889779, −1.54198409796469933125889280771, 0.934601539377809256200566145486, 2.67648847348555367910713459262, 3.74528580302142896952280551722, 4.41737117186183623498957868480, 5.62850406607789647041136647070, 6.96298968040299291576002506768, 8.646514318668455956841171675829, 9.308660888356780530436386788099, 9.985320105309470377632833256166, 10.78329397926811855991013299991

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