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Properties

Label 2-1530-17.4-c1-0-11
Degree $2$
Conductor $1530$
Sign $0.988 + 0.151i$
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  − i·2-s − 4-s + (0.707 − 0.707i)5-s + (2.41 + 2.41i)7-s + i·8-s + (−0.707 − 0.707i)10-s + (−1 − i)11-s + 13-s + (2.41 − 2.41i)14-s + 16-s + (−0.121 + 4.12i)17-s − 0.414i·19-s + (−0.707 + 0.707i)20-s + (−1 + i)22-s + (6.24 + 6.24i)23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.316 − 0.316i)5-s + (0.912 + 0.912i)7-s + 0.353i·8-s + (−0.223 − 0.223i)10-s + (−0.301 − 0.301i)11-s + 0.277·13-s + (0.645 − 0.645i)14-s + 0.250·16-s + (−0.0294 + 0.999i)17-s − 0.0950i·19-s + (−0.158 + 0.158i)20-s + (−0.213 + 0.213i)22-s + (1.30 + 1.30i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1530\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $0.988 + 0.151i$
Analytic conductor: \(12.2171\)
Root analytic conductor: \(3.49529\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1530} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1530,\ (\ :1/2),\ 0.988 + 0.151i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.842143845\)
\(L(\frac12)\) \(\approx\) \(1.842143845\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
5 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 + (0.121 - 4.12i)T \)
good7 \( 1 + (-2.41 - 2.41i)T + 7iT^{2} \)
11 \( 1 + (1 + i)T + 11iT^{2} \)
13 \( 1 - T + 13T^{2} \)
19 \( 1 + 0.414iT - 19T^{2} \)
23 \( 1 + (-6.24 - 6.24i)T + 23iT^{2} \)
29 \( 1 + (2.94 - 2.94i)T - 29iT^{2} \)
31 \( 1 + (2.29 - 2.29i)T - 31iT^{2} \)
37 \( 1 + (-4.41 + 4.41i)T - 37iT^{2} \)
41 \( 1 + (4.65 + 4.65i)T + 41iT^{2} \)
43 \( 1 + 1.75iT - 43T^{2} \)
47 \( 1 - 5.24T + 47T^{2} \)
53 \( 1 - 13.4iT - 53T^{2} \)
59 \( 1 - 8.89iT - 59T^{2} \)
61 \( 1 + (-9.77 - 9.77i)T + 61iT^{2} \)
67 \( 1 - 3.17T + 67T^{2} \)
71 \( 1 + (-9.70 + 9.70i)T - 71iT^{2} \)
73 \( 1 + (1.53 - 1.53i)T - 73iT^{2} \)
79 \( 1 + (0.242 + 0.242i)T + 79iT^{2} \)
83 \( 1 + 9.89iT - 83T^{2} \)
89 \( 1 + 3.34T + 89T^{2} \)
97 \( 1 + (-5.87 + 5.87i)T - 97iT^{2} \)
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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

−9.184437205030219351853706843347, −8.917525275941149356648624233318, −8.097326936637744871172006682753, −7.16238315507112008743976275161, −5.67519453672985837579342802428, −5.46090944960160389138985470397, −4.35206798138933359547290899755, −3.28466548162264318826761228933, −2.16976412263466744435582224265, −1.28101579889472545220028959238, 0.839619131026778047371248417849, 2.33337225233744171720050748419, 3.65328477199850699496063980267, 4.72430060228352333709364674757, 5.20266924380257069645990217008, 6.46227725117606588961557945534, 7.02113743987566776688020432220, 7.83256951975652790962226889199, 8.447214262499219613336313854540, 9.504174579766163252679628323666

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