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Properties

Label 2-30-5.4-c11-0-6
Degree $2$
Conductor $30$
Sign $0.716 + 0.697i$
Analytic cond. $23.0502$
Root an. cond. $4.80107$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 32i·2-s + 243i·3-s − 1.02e3·4-s + (−5.00e3 − 4.87e3i)5-s − 7.77e3·6-s + 7.87e4i·7-s − 3.27e4i·8-s − 5.90e4·9-s + (1.55e5 − 1.60e5i)10-s − 4.93e5·11-s − 2.48e5i·12-s − 2.35e6i·13-s − 2.52e6·14-s + (1.18e6 − 1.21e6i)15-s + 1.04e6·16-s + 4.45e6i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (−0.716 − 0.697i)5-s − 0.408·6-s + 1.77i·7-s − 0.353i·8-s − 0.333·9-s + (0.493 − 0.506i)10-s − 0.924·11-s − 0.288i·12-s − 1.75i·13-s − 1.25·14-s + (0.402 − 0.413i)15-s + 0.250·16-s + 0.760i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30\)    =    \(2 \cdot 3 \cdot 5\)
Sign: $0.716 + 0.697i$
Analytic conductor: \(23.0502\)
Root analytic conductor: \(4.80107\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{30} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 30,\ (\ :11/2),\ 0.716 + 0.697i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.542396 - 0.220324i\)
\(L(\frac12)\) \(\approx\) \(0.542396 - 0.220324i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 32iT \)
3 \( 1 - 243iT \)
5 \( 1 + (5.00e3 + 4.87e3i)T \)
good7 \( 1 - 7.87e4iT - 1.97e9T^{2} \)
11 \( 1 + 4.93e5T + 2.85e11T^{2} \)
13 \( 1 + 2.35e6iT - 1.79e12T^{2} \)
17 \( 1 - 4.45e6iT - 3.42e13T^{2} \)
19 \( 1 - 9.21e6T + 1.16e14T^{2} \)
23 \( 1 + 5.33e7iT - 9.52e14T^{2} \)
29 \( 1 - 6.06e7T + 1.22e16T^{2} \)
31 \( 1 - 1.54e8T + 2.54e16T^{2} \)
37 \( 1 + 3.40e8iT - 1.77e17T^{2} \)
41 \( 1 + 6.07e8T + 5.50e17T^{2} \)
43 \( 1 - 4.12e8iT - 9.29e17T^{2} \)
47 \( 1 + 2.74e9iT - 2.47e18T^{2} \)
53 \( 1 + 9.65e8iT - 9.26e18T^{2} \)
59 \( 1 + 2.66e9T + 3.01e19T^{2} \)
61 \( 1 + 5.86e9T + 4.35e19T^{2} \)
67 \( 1 - 8.92e9iT - 1.22e20T^{2} \)
71 \( 1 + 1.04e10T + 2.31e20T^{2} \)
73 \( 1 + 1.69e10iT - 3.13e20T^{2} \)
79 \( 1 + 2.77e10T + 7.47e20T^{2} \)
83 \( 1 + 5.41e10iT - 1.28e21T^{2} \)
89 \( 1 + 3.53e10T + 2.77e21T^{2} \)
97 \( 1 - 1.08e11iT - 7.15e21T^{2} \)
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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

−14.97030196125755699443342879377, −12.95179128104536404361144007175, −12.07816618657155230015234234388, −10.29601725256030845313515611191, −8.696655729379064812630857416096, −8.082201106299546118512740973930, −5.76790943958575966796842659975, −4.91196069549311910721823771022, −2.95858675657909595509156928211, −0.22460499025096899667949478958, 1.23802608673735608325024454052, 3.11070899272506628444719819268, 4.45834585138622326845891430848, 6.93222580675907995527927061340, 7.77055581704314317493297817599, 9.795199531052564223665529829692, 11.05107008284676020801455515243, 11.85586437236607423984659298004, 13.65036174004387332765921910769, 13.97607279433017626419909952953

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