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Properties

Label 2-30-15.2-c9-0-3
Degree $2$
Conductor $30$
Sign $0.365 - 0.930i$
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 + (−125. + 62.4i)3-s − 256. i·4-s + (−1.15e3 − 792. i)5-s + (715. − 2.12e3i)6-s + (−1.31e3 − 1.31e3i)7-s + (2.89e3 + 2.89e3i)8-s + (1.18e4 − 1.56e4i)9-s + (2.19e4 − 4.05e3i)10-s − 5.74e4i·11-s + (1.59e4 + 3.21e4i)12-s + (−1.01e5 + 1.01e5i)13-s + 2.97e4·14-s + (1.94e5 + 2.77e4i)15-s − 6.55e4·16-s + (2.89e4 − 2.89e4i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.895 + 0.444i)3-s − 0.500i·4-s + (−0.823 − 0.567i)5-s + (0.225 − 0.670i)6-s + (−0.206 − 0.206i)7-s + (0.250 + 0.250i)8-s + (0.604 − 0.796i)9-s + (0.695 − 0.128i)10-s − 1.18i·11-s + (0.222 + 0.447i)12-s + (−0.988 + 0.988i)13-s + 0.206·14-s + (0.989 + 0.141i)15-s − 0.250·16-s + (0.0840 − 0.0840i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.365 - 0.930i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.365 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30\)    =    \(2 \cdot 3 \cdot 5\)
Sign: $0.365 - 0.930i$
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: \((2,\ 30,\ (\ :9/2),\ 0.365 - 0.930i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.484893 + 0.330688i\)
\(L(\frac12)\) \(\approx\) \(0.484893 + 0.330688i\)
\(L(\frac{11}{2})\) 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 \( 1 + (125. - 62.4i)T \)
5 \( 1 + (1.15e3 + 792. i)T \)
good7 \( 1 + (1.31e3 + 1.31e3i)T + 4.03e7iT^{2} \)
11 \( 1 + 5.74e4iT - 2.35e9T^{2} \)
13 \( 1 + (1.01e5 - 1.01e5i)T - 1.06e10iT^{2} \)
17 \( 1 + (-2.89e4 + 2.89e4i)T - 1.18e11iT^{2} \)
19 \( 1 - 7.41e5iT - 3.22e11T^{2} \)
23 \( 1 + (-9.75e4 - 9.75e4i)T + 1.80e12iT^{2} \)
29 \( 1 - 4.65e6T + 1.45e13T^{2} \)
31 \( 1 + 2.20e6T + 2.64e13T^{2} \)
37 \( 1 + (-4.81e6 - 4.81e6i)T + 1.29e14iT^{2} \)
41 \( 1 + 1.92e7iT - 3.27e14T^{2} \)
43 \( 1 + (-2.22e7 + 2.22e7i)T - 5.02e14iT^{2} \)
47 \( 1 + (1.46e7 - 1.46e7i)T - 1.11e15iT^{2} \)
53 \( 1 + (-6.77e7 - 6.77e7i)T + 3.29e15iT^{2} \)
59 \( 1 + 5.74e7T + 8.66e15T^{2} \)
61 \( 1 - 1.76e8T + 1.16e16T^{2} \)
67 \( 1 + (-1.92e8 - 1.92e8i)T + 2.72e16iT^{2} \)
71 \( 1 - 1.21e8iT - 4.58e16T^{2} \)
73 \( 1 + (5.63e7 - 5.63e7i)T - 5.88e16iT^{2} \)
79 \( 1 - 5.37e8iT - 1.19e17T^{2} \)
83 \( 1 + (9.18e7 + 9.18e7i)T + 1.86e17iT^{2} \)
89 \( 1 + 4.92e8T + 3.50e17T^{2} \)
97 \( 1 + (8.26e8 + 8.26e8i)T + 7.60e17iT^{2} \)
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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

−15.66967431128279289174053074631, −14.23752516509113049735396864014, −12.36507243414792244873039881712, −11.35195008038240049084527296437, −9.998144776502013349432207771655, −8.625768762772555452851493426220, −7.07272691712193433114446358844, −5.55157812760354723524298471269, −4.06591800167716376264868735746, −0.789805676046603090837037207332, 0.48820401689420821640168034968, 2.57578091014998966569763338445, 4.72352772728025283868861279698, 6.83499175615023654031741841455, 7.82210832773708947898708352301, 9.870171010007778024321229962617, 10.98256509527294761890629760885, 12.09433825627711202630910116332, 12.88475251582455985636189759728, 14.94889843510252193944594867422

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