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Properties

Label 2-1530-15.2-c1-0-13
Degree $2$
Conductor $1530$
Sign $0.960 - 0.279i$
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 + (−1.73 + 1.41i)5-s + (2 + 2i)7-s + (0.707 + 0.707i)8-s + (0.224 − 2.22i)10-s − 2.82i·11-s + (2.44 − 2.44i)13-s − 2.82·14-s − 1.00·16-s + (0.707 − 0.707i)17-s − 2.89i·19-s + (1.41 + 1.73i)20-s + (2.00 + 2.00i)22-s + (−1.73 − 1.73i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.774 + 0.632i)5-s + (0.755 + 0.755i)7-s + (0.250 + 0.250i)8-s + (0.0710 − 0.703i)10-s − 0.852i·11-s + (0.679 − 0.679i)13-s − 0.755·14-s − 0.250·16-s + (0.171 − 0.171i)17-s − 0.665i·19-s + (0.316 + 0.387i)20-s + (0.426 + 0.426i)22-s + (−0.361 − 0.361i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.279i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 - 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1530\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $0.960 - 0.279i$
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: \((2,\ 1530,\ (\ :1/2),\ 0.960 - 0.279i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.193398586\)
\(L(\frac12)\) \(\approx\) \(1.193398586\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 - 1.41i)T \)
17 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + (-2 - 2i)T + 7iT^{2} \)
11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 + (-2.44 + 2.44i)T - 13iT^{2} \)
19 \( 1 + 2.89iT - 19T^{2} \)
23 \( 1 + (1.73 + 1.73i)T + 23iT^{2} \)
29 \( 1 + 2.82T + 29T^{2} \)
31 \( 1 - 7.34T + 31T^{2} \)
37 \( 1 + (-1.55 - 1.55i)T + 37iT^{2} \)
41 \( 1 + 6.29iT - 41T^{2} \)
43 \( 1 + (1.44 - 1.44i)T - 43iT^{2} \)
47 \( 1 + (-2.82 + 2.82i)T - 47iT^{2} \)
53 \( 1 + (-8.34 - 8.34i)T + 53iT^{2} \)
59 \( 1 + 2.04T + 59T^{2} \)
61 \( 1 + 9.34T + 61T^{2} \)
67 \( 1 + (-3.44 - 3.44i)T + 67iT^{2} \)
71 \( 1 - 6.29iT - 71T^{2} \)
73 \( 1 + (-11.3 + 11.3i)T - 73iT^{2} \)
79 \( 1 - 1.55iT - 79T^{2} \)
83 \( 1 + (-2.82 - 2.82i)T + 83iT^{2} \)
89 \( 1 - 15.2T + 89T^{2} \)
97 \( 1 + (-1.55 - 1.55i)T + 97iT^{2} \)
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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

−9.245921114719451816233205251224, −8.441569524968942488057342545767, −8.089103398265751882817221320486, −7.21809049331057841930712805947, −6.27142590549344164639979642321, −5.58783650070710212529703173456, −4.57664049135775748259926632725, −3.40917877650911699986389440529, −2.38968430388368408222338118501, −0.73375296841782851400552388412, 1.01982285990240310877031492289, 1.93272067708930418358105668801, 3.56251553156321246836458111030, 4.24407060230675537678889361794, 4.93890699997963741631699100384, 6.31255659717371847125384687904, 7.40494241833875642116077650943, 7.88424197037009139244519317202, 8.579553789099379125375128093835, 9.441770054116485488801301186110

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