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Properties

Label 2-30-5.4-c1-0-0
Degree $2$
Conductor $30$
Sign $0.894 - 0.447i$
Analytic cond. $0.239551$
Root an. cond. $0.489439$
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  + i·2-s − i·3-s − 4-s + (−2 + i)5-s + 6-s − 2i·7-s − i·8-s − 9-s + (−1 − 2i)10-s + 2·11-s + i·12-s + 6i·13-s + 2·14-s + (1 + 2i)15-s + 16-s − 2i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (−0.894 + 0.447i)5-s + 0.408·6-s − 0.755i·7-s − 0.353i·8-s − 0.333·9-s + (−0.316 − 0.632i)10-s + 0.603·11-s + 0.288i·12-s + 1.66i·13-s + 0.534·14-s + (0.258 + 0.516i)15-s + 0.250·16-s − 0.485i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30\)    =    \(2 \cdot 3 \cdot 5\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(0.239551\)
Root analytic conductor: \(0.489439\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{30} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 30,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.612627 + 0.144621i\)
\(L(\frac12)\) \(\approx\) \(0.612627 + 0.144621i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + iT \)
5 \( 1 + (2 - i)T \)
good7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 10T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−16.96826441731702099101614986322, −16.14791334000609191876902356433, −14.60665231932445477324661417962, −13.88854108804486098115315613885, −12.26964569530337323975039188135, −11.03406984828253276966084497277, −9.062236764009944146012759381636, −7.48664995322902406921466101924, −6.63650189600152426497619539048, −4.16710137136039276355392879535, 3.56141683746843219707232773454, 5.35115232455633761080372105825, 8.085457671479813666280641047905, 9.291364285307452496699123458412, 10.79733812128479603885789976626, 11.95451807699588727491650213315, 12.96250512409104260660029756949, 14.82613947152933352288907316812, 15.63832276156840719790648214054, 17.01977398561147585569982166153

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