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Properties

Label 2-30-15.2-c9-0-11
Degree $2$
Conductor $30$
Sign $-0.0494 + 0.998i$
Analytic cond. $15.4510$
Root an. cond. $3.93078$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (11.3 − 11.3i)2-s + (−19.5 + 138. i)3-s − 256. i·4-s + (−442. − 1.32e3i)5-s + (1.35e3 + 1.79e3i)6-s + (4.22e3 + 4.22e3i)7-s + (−2.89e3 − 2.89e3i)8-s + (−1.89e4 − 5.43e3i)9-s + (−2.00e4 − 9.99e3i)10-s − 6.24e4i·11-s + (3.55e4 + 5.00e3i)12-s + (1.80e4 − 1.80e4i)13-s + 9.56e4·14-s + (1.92e5 − 3.55e4i)15-s − 6.55e4·16-s + (3.63e5 − 3.63e5i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.139 + 0.990i)3-s − 0.500i·4-s + (−0.316 − 0.948i)5-s + (0.425 + 0.564i)6-s + (0.665 + 0.665i)7-s + (−0.250 − 0.250i)8-s + (−0.961 − 0.275i)9-s + (−0.632 − 0.315i)10-s − 1.28i·11-s + (0.495 + 0.0696i)12-s + (0.175 − 0.175i)13-s + 0.665·14-s + (0.983 − 0.181i)15-s − 0.250·16-s + (1.05 − 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0494 + 0.998i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.0494 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30\)    =    \(2 \cdot 3 \cdot 5\)
Sign: $-0.0494 + 0.998i$
Analytic conductor: \(15.4510\)
Root analytic conductor: \(3.93078\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{30} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 30,\ (\ :9/2),\ -0.0494 + 0.998i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.30464 - 1.37083i\)
\(L(\frac12)\) \(\approx\) \(1.30464 - 1.37083i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-11.3 + 11.3i)T \)
3 \( 1 + (19.5 - 138. i)T \)
5 \( 1 + (442. + 1.32e3i)T \)
good7 \( 1 + (-4.22e3 - 4.22e3i)T + 4.03e7iT^{2} \)
11 \( 1 + 6.24e4iT - 2.35e9T^{2} \)
13 \( 1 + (-1.80e4 + 1.80e4i)T - 1.06e10iT^{2} \)
17 \( 1 + (-3.63e5 + 3.63e5i)T - 1.18e11iT^{2} \)
19 \( 1 + 9.79e5iT - 3.22e11T^{2} \)
23 \( 1 + (2.60e5 + 2.60e5i)T + 1.80e12iT^{2} \)
29 \( 1 - 1.27e6T + 1.45e13T^{2} \)
31 \( 1 + 6.88e6T + 2.64e13T^{2} \)
37 \( 1 + (-1.46e7 - 1.46e7i)T + 1.29e14iT^{2} \)
41 \( 1 - 3.70e6iT - 3.27e14T^{2} \)
43 \( 1 + (5.49e6 - 5.49e6i)T - 5.02e14iT^{2} \)
47 \( 1 + (1.61e7 - 1.61e7i)T - 1.11e15iT^{2} \)
53 \( 1 + (-2.23e6 - 2.23e6i)T + 3.29e15iT^{2} \)
59 \( 1 - 1.50e8T + 8.66e15T^{2} \)
61 \( 1 + 3.47e7T + 1.16e16T^{2} \)
67 \( 1 + (1.76e8 + 1.76e8i)T + 2.72e16iT^{2} \)
71 \( 1 - 4.39e7iT - 4.58e16T^{2} \)
73 \( 1 + (1.17e8 - 1.17e8i)T - 5.88e16iT^{2} \)
79 \( 1 + 5.08e6iT - 1.19e17T^{2} \)
83 \( 1 + (-1.10e8 - 1.10e8i)T + 1.86e17iT^{2} \)
89 \( 1 + 3.73e8T + 3.50e17T^{2} \)
97 \( 1 + (-5.17e8 - 5.17e8i)T + 7.60e17iT^{2} \)
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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

−14.67756752194833547219766290212, −13.38438367511279033147188427994, −11.80190152642581466810235116517, −11.17046450203583869502792423943, −9.447248328147675297337576125338, −8.414237774439749222761761985628, −5.60292416352431892292865281479, −4.71194661034798062737283476159, −3.06758397930581695827305235107, −0.67222279254477016975953185876, 1.79924130853802699110250050116, 3.84703503873254816560734050343, 5.87436073376251704965679561343, 7.27162388140060938813827403833, 7.923971245363949655210444743046, 10.41096090372367200400147774492, 11.80060766447820299502734862538, 12.82821240472228115526980350064, 14.39379975090235428092645426214, 14.67369481028152164321006425148

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