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Properties

Label 2-483-7.2-c1-0-21
Degree $2$
Conductor $483$
Sign $0.943 - 0.331i$
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  + (−1.13 + 1.97i)2-s + (0.5 + 0.866i)3-s + (−1.58 − 2.75i)4-s + (0.619 − 1.07i)5-s − 2.27·6-s + (1.84 − 1.89i)7-s + 2.67·8-s + (−0.499 + 0.866i)9-s + (1.40 + 2.44i)10-s + (−3.06 − 5.30i)11-s + (1.58 − 2.75i)12-s + 0.910·13-s + (1.62 + 5.79i)14-s + 1.23·15-s + (0.133 − 0.230i)16-s + (−2.45 − 4.25i)17-s + ⋯
L(s)  = 1  + (−0.804 + 1.39i)2-s + (0.288 + 0.499i)3-s + (−0.793 − 1.37i)4-s + (0.276 − 0.479i)5-s − 0.928·6-s + (0.698 − 0.715i)7-s + 0.945·8-s + (−0.166 + 0.288i)9-s + (0.445 + 0.771i)10-s + (−0.923 − 1.59i)11-s + (0.458 − 0.793i)12-s + 0.252·13-s + (0.435 + 1.54i)14-s + 0.319·15-s + (0.0332 − 0.0576i)16-s + (−0.595 − 1.03i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.943 - 0.331i$
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: \((2,\ 483,\ (\ :1/2),\ 0.943 - 0.331i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.912742 + 0.155528i\)
\(L(\frac12)\) \(\approx\) \(0.912742 + 0.155528i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.84 + 1.89i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (1.13 - 1.97i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-0.619 + 1.07i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.06 + 5.30i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.910T + 13T^{2} \)
17 \( 1 + (2.45 + 4.25i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.45 + 5.97i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 - 3.36T + 29T^{2} \)
31 \( 1 + (2.71 + 4.70i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.27 - 5.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.52T + 41T^{2} \)
43 \( 1 + 0.231T + 43T^{2} \)
47 \( 1 + (6.67 - 11.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.11 + 3.66i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.377 - 0.653i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.78 - 3.09i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.24 - 3.89i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 15.8T + 71T^{2} \)
73 \( 1 + (-2.13 - 3.70i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.111 + 0.192i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 15.5T + 83T^{2} \)
89 \( 1 + (0.279 - 0.484i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 17.6T + 97T^{2} \)
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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

−10.87674949832395887603392166898, −9.731199035061476945127223829862, −9.009556320694420070043432212360, −8.299043763954223245428910930101, −7.60791537489116359468999067078, −6.59139294368450922257029386956, −5.35136711017361938773845579948, −4.85810924888149485958643179225, −3.08665792540709661096599565806, −0.72448161224095031829397617916, 1.77611766155768946216789031613, 2.26752791881068973263659267807, 3.57972404695650391187486829735, 5.07810688908628795654145364306, 6.45925796393490559434377436617, 7.75541123371118661148372653490, 8.370963852932171477479263730524, 9.307859078318677633660290158565, 10.30060553620046080081444764675, 10.65149150194121711042768694515

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