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Properties

Label 2-1530-255.98-c1-0-20
Degree $2$
Conductor $1530$
Sign $0.186 + 0.982i$
Analytic cond. $12.2171$
Root an. cond. $3.49529$
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  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (−2.18 + 0.482i)5-s − 3.29i·7-s + (0.707 + 0.707i)8-s + (1.20 − 1.88i)10-s + (2.44 + 2.44i)11-s + (−0.00346 − 0.00346i)13-s + (2.32 + 2.32i)14-s − 1.00·16-s + (4.02 + 0.875i)17-s − 5.57·19-s + (0.482 + 2.18i)20-s − 3.46·22-s + 0.449i·23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.976 + 0.215i)5-s − 1.24i·7-s + (0.250 + 0.250i)8-s + (0.380 − 0.596i)10-s + (0.738 + 0.738i)11-s + (−0.000959 − 0.000959i)13-s + (0.621 + 0.621i)14-s − 0.250·16-s + (0.977 + 0.212i)17-s − 1.27·19-s + (0.107 + 0.488i)20-s − 0.738·22-s + 0.0936i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.186 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1530\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $0.186 + 0.982i$
Analytic conductor: \(12.2171\)
Root analytic conductor: \(3.49529\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1530} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1530,\ (\ :1/2),\ 0.186 + 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6855023183\)
\(L(\frac12)\) \(\approx\) \(0.6855023183\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 \)
5 \( 1 + (2.18 - 0.482i)T \)
17 \( 1 + (-4.02 - 0.875i)T \)
good7 \( 1 + 3.29iT - 7T^{2} \)
11 \( 1 + (-2.44 - 2.44i)T + 11iT^{2} \)
13 \( 1 + (0.00346 + 0.00346i)T + 13iT^{2} \)
19 \( 1 + 5.57T + 19T^{2} \)
23 \( 1 - 0.449iT - 23T^{2} \)
29 \( 1 + (0.747 + 0.747i)T + 29iT^{2} \)
31 \( 1 + (0.254 + 0.254i)T + 31iT^{2} \)
37 \( 1 + 3.06T + 37T^{2} \)
41 \( 1 + (4.13 + 4.13i)T + 41iT^{2} \)
43 \( 1 + (-2.34 + 2.34i)T - 43iT^{2} \)
47 \( 1 + (2.11 + 2.11i)T + 47iT^{2} \)
53 \( 1 + (7.60 + 7.60i)T + 53iT^{2} \)
59 \( 1 + 7.62iT - 59T^{2} \)
61 \( 1 + (-3.72 + 3.72i)T - 61iT^{2} \)
67 \( 1 + (-8.05 + 8.05i)T - 67iT^{2} \)
71 \( 1 + (2.23 - 2.23i)T - 71iT^{2} \)
73 \( 1 + 10.7iT - 73T^{2} \)
79 \( 1 + (10.1 + 10.1i)T + 79iT^{2} \)
83 \( 1 + (-5.69 - 5.69i)T + 83iT^{2} \)
89 \( 1 + 7.62T + 89T^{2} \)
97 \( 1 - 11.8T + 97T^{2} \)
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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

−9.225669636000848139058669817575, −8.262533017130019581615931270472, −7.69596612491791532751376171321, −6.93906730531642720498296808769, −6.44942556382669373717081942994, −5.05388631470471703765482117543, −4.15033593335210462961670719345, −3.51051485947405166761700494796, −1.74417818972591057283179243346, −0.36145517653933930510723851123, 1.22338875666681752023611429228, 2.62092258451180342918653903475, 3.49436500865697663593532484426, 4.42217835381883138856124517628, 5.53491330021858073584097423484, 6.43772526790720003453502183588, 7.45109468365929122105021418171, 8.427489555629990053182463612865, 8.666222778791661492829108132572, 9.486393218358479660767332062092

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