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Properties

Label 2-1530-15.2-c1-0-12
Degree $2$
Conductor $1530$
Sign $0.749 + 0.662i$
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 + (−2.12 − 0.707i)5-s + (−1 − i)7-s + (0.707 + 0.707i)8-s + (2 − 0.999i)10-s + 5.65i·11-s + 1.41·14-s − 1.00·16-s + (0.707 − 0.707i)17-s − 4i·19-s + (−0.707 + 2.12i)20-s + (−4.00 − 4.00i)22-s + (3.99 + 3i)25-s + (−1.00 + 1.00i)28-s + 1.41·29-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.948 − 0.316i)5-s + (−0.377 − 0.377i)7-s + (0.250 + 0.250i)8-s + (0.632 − 0.316i)10-s + 1.70i·11-s + 0.377·14-s − 0.250·16-s + (0.171 − 0.171i)17-s − 0.917i·19-s + (−0.158 + 0.474i)20-s + (−0.852 − 0.852i)22-s + (0.799 + 0.600i)25-s + (−0.188 + 0.188i)28-s + 0.262·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1530\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $0.749 + 0.662i$
Analytic conductor: \(12.2171\)
Root analytic conductor: \(3.49529\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1530} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1530,\ (\ :1/2),\ 0.749 + 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7313584850\)
\(L(\frac12)\) \(\approx\) \(0.7313584850\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.12 + 0.707i)T \)
17 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + (1 + i)T + 7iT^{2} \)
11 \( 1 - 5.65iT - 11T^{2} \)
13 \( 1 - 13iT^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + 23iT^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + (-1 - i)T + 37iT^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 + (-7 + 7i)T - 43iT^{2} \)
47 \( 1 + (-2.82 + 2.82i)T - 47iT^{2} \)
53 \( 1 + (7.07 + 7.07i)T + 53iT^{2} \)
59 \( 1 - 1.41T + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 + (5 + 5i)T + 67iT^{2} \)
71 \( 1 + 9.89iT - 71T^{2} \)
73 \( 1 + (-10 + 10i)T - 73iT^{2} \)
79 \( 1 + 8iT - 79T^{2} \)
83 \( 1 + (1.41 + 1.41i)T + 83iT^{2} \)
89 \( 1 - 1.41T + 89T^{2} \)
97 \( 1 + (-4 - 4i)T + 97iT^{2} \)
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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

−9.307706608213797693451678753034, −8.572832086939430873272564702524, −7.47453271726463794075305255385, −7.30068417466814138996738039711, −6.39571071471323559189155435993, −5.06778389510876232399491319964, −4.50734254921392764120915115438, −3.44494911793915598334218731596, −1.98291576337239375673385072303, −0.44351355740583283098420303783, 0.955848957334250073364273180703, 2.62989182948122126116163397898, 3.42956300122998415919846469468, 4.12036056564128352816723569555, 5.57982639890345905856997664921, 6.31048897383123393196892433403, 7.40927859948473105377340126533, 8.091342595427514494968477634520, 8.733470989441587843176138962979, 9.464249530799597046000019195784

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