Properties

Label 2-483-7.4-c1-0-23
Degree $2$
Conductor $483$
Sign $-0.594 + 0.804i$
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  + (−0.144 − 0.251i)2-s + (0.5 − 0.866i)3-s + (0.957 − 1.65i)4-s + (−1.16 − 2.01i)5-s − 0.289·6-s + (2.35 − 1.20i)7-s − 1.13·8-s + (−0.499 − 0.866i)9-s + (−0.337 + 0.584i)10-s + (−0.365 + 0.633i)11-s + (−0.957 − 1.65i)12-s − 0.115·13-s + (−0.644 − 0.415i)14-s − 2.32·15-s + (−1.75 − 3.03i)16-s + (−0.394 + 0.683i)17-s + ⋯
L(s)  = 1  + (−0.102 − 0.177i)2-s + (0.288 − 0.499i)3-s + (0.478 − 0.829i)4-s + (−0.520 − 0.900i)5-s − 0.118·6-s + (0.889 − 0.456i)7-s − 0.401·8-s + (−0.166 − 0.288i)9-s + (−0.106 + 0.184i)10-s + (−0.110 + 0.190i)11-s + (−0.276 − 0.478i)12-s − 0.0319·13-s + (−0.172 − 0.111i)14-s − 0.600·15-s + (−0.437 − 0.758i)16-s + (−0.0956 + 0.165i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.594 + 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.594 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.594 + 0.804i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.594 + 0.804i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.692433 - 1.37204i\)
\(L(\frac12)\) \(\approx\) \(0.692433 - 1.37204i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.35 + 1.20i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (0.144 + 0.251i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.16 + 2.01i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.365 - 0.633i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.115T + 13T^{2} \)
17 \( 1 + (0.394 - 0.683i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.46 - 5.99i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 + 6.26T + 29T^{2} \)
31 \( 1 + (0.747 - 1.29i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.07 + 3.58i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.20T + 41T^{2} \)
43 \( 1 - 1.50T + 43T^{2} \)
47 \( 1 + (-1.89 - 3.28i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.40 + 5.90i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.09 + 5.35i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.19 - 2.07i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.18 + 7.25i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.53T + 71T^{2} \)
73 \( 1 + (4.95 - 8.57i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.241 + 0.419i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.84T + 83T^{2} \)
89 \( 1 + (1.53 + 2.65i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 14.8T + 97T^{2} \)
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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

−10.77233910669579233470450501461, −9.787041490894708379543974108716, −8.839548897725342786777243323834, −7.87425752504670799702557893839, −7.24802133616272752463079227407, −5.88798133389985753014895233612, −5.02118598139517760877859211724, −3.79911245489724822720200784236, −2.02349503920634885577049846436, −0.988473831032917368374634401854, 2.43248964052498885208705690508, 3.27034053515498270791376095632, 4.42604076784451433152400194981, 5.69064736650993407166536496570, 7.09167059316652713928892535582, 7.56075724814783615962715982656, 8.559638852601304179574347096267, 9.292764041250214918869184227650, 10.69910204275552460906669649549, 11.31160177408830454436747899327

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