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Properties

Label 2-483-7.2-c1-0-18
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  + (−0.5 − 0.866i)3-s + (1 + 1.73i)4-s + (2.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + (−3 − 5.19i)11-s + (0.999 − 1.73i)12-s + 5·13-s + (−1.99 + 3.46i)16-s + (3 + 5.19i)17-s + (0.5 − 0.866i)19-s + (−2 − 1.73i)21-s + (0.5 − 0.866i)23-s + (2.5 + 4.33i)25-s + 0.999·27-s + (4 + 3.46i)28-s + 6·29-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.5 + 0.866i)4-s + (0.944 − 0.327i)7-s + (−0.166 + 0.288i)9-s + (−0.904 − 1.56i)11-s + (0.288 − 0.499i)12-s + 1.38·13-s + (−0.499 + 0.866i)16-s + (0.727 + 1.26i)17-s + (0.114 − 0.198i)19-s + (−0.436 − 0.377i)21-s + (0.104 − 0.180i)23-s + (0.5 + 0.866i)25-s + 0.192·27-s + (0.755 + 0.654i)28-s + 1.11·29-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

\(L(1)\) \(\approx\) \(1.60502 - 0.101855i\)
\(L(\frac12)\) \(\approx\) \(1.60502 - 0.101855i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6 + 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3 - 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7 - 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.5 - 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−11.04874881047402161388100752147, −10.59552810844395674561022520822, −8.666904667437368768231839564594, −8.215957292124443851724440526804, −7.51641584421071846187434081491, −6.29181313736809311815566810704, −5.53684978854764559278433097917, −3.98305005149303116711957671665, −2.93954750076820013256659025927, −1.33050985624522978346679624229, 1.41228084486232279061352243294, 2.81032073272351323418807827555, 4.67814758332579250034171209939, 5.14138898611262881020016876981, 6.23549363334136228540821825558, 7.28709521874879857386142411812, 8.310238398719040413340336024257, 9.473331339187304138948343335740, 10.24188879175280374370204475737, 10.87699742890481836393325049772

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