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Properties

Label 2-483-21.5-c1-0-6
Degree $2$
Conductor $483$
Sign $0.840 - 0.541i$
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  + (−1.07 − 0.619i)2-s + (−0.722 − 1.57i)3-s + (−0.231 − 0.401i)4-s + (−0.273 + 0.474i)5-s + (−0.199 + 2.13i)6-s + (2.55 − 0.700i)7-s + 3.05i·8-s + (−1.95 + 2.27i)9-s + (0.587 − 0.339i)10-s + (−3.30 + 1.90i)11-s + (−0.464 + 0.655i)12-s + 6.33i·13-s + (−3.17 − 0.828i)14-s + (0.944 + 0.0882i)15-s + (1.42 − 2.47i)16-s + (0.532 + 0.922i)17-s + ⋯
L(s)  = 1  + (−0.758 − 0.438i)2-s + (−0.417 − 0.908i)3-s + (−0.115 − 0.200i)4-s + (−0.122 + 0.212i)5-s + (−0.0815 + 0.872i)6-s + (0.964 − 0.264i)7-s + 1.07i·8-s + (−0.651 + 0.758i)9-s + (0.185 − 0.107i)10-s + (−0.995 + 0.574i)11-s + (−0.134 + 0.189i)12-s + 1.75i·13-s + (−0.847 − 0.221i)14-s + (0.243 + 0.0227i)15-s + (0.357 − 0.618i)16-s + (0.129 + 0.223i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 - 0.541i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.840 - 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.840 - 0.541i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.840 - 0.541i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.452909 + 0.133129i\)
\(L(\frac12)\) \(\approx\) \(0.452909 + 0.133129i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.722 + 1.57i)T \)
7 \( 1 + (-2.55 + 0.700i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
good2 \( 1 + (1.07 + 0.619i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (0.273 - 0.474i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.30 - 1.90i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 6.33iT - 13T^{2} \)
17 \( 1 + (-0.532 - 0.922i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.98 + 2.30i)T + (9.5 + 16.4i)T^{2} \)
29 \( 1 - 4.33iT - 29T^{2} \)
31 \( 1 + (2.78 - 1.60i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.00 + 1.74i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 7.04T + 41T^{2} \)
43 \( 1 + 7.94T + 43T^{2} \)
47 \( 1 + (1.88 - 3.26i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.11 + 0.644i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.05 + 3.56i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.33 - 2.50i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.39 + 7.61i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.0iT - 71T^{2} \)
73 \( 1 + (9.71 - 5.60i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.987 + 1.71i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.31T + 83T^{2} \)
89 \( 1 + (6.24 - 10.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.90iT - 97T^{2} \)
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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

−11.04888803599610571562589228825, −10.44706165949065241121817922880, −9.197524674014631291787280638702, −8.407285916460607573887929055174, −7.51790136763662900999860121382, −6.70009095435212638742479476815, −5.35857596194329463382145814883, −4.54689198177206834856274258133, −2.32301450895430834251032852030, −1.49561115481771806580917052584, 0.40182453819169454584396980732, 2.97696607997108042807774386636, 4.24377213610021246224053910072, 5.27154377120873297905592660549, 6.10817523348978398800221271743, 7.74671750604399949075380812206, 8.214089858221867625691142264709, 8.915035719787258531615750830511, 10.17700002757219257974111780077, 10.51449905995614486694147847326

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