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Properties

Label 2-30-5.4-c9-0-7
Degree $2$
Conductor $30$
Sign $-0.474 + 0.880i$
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  − 16i·2-s + 81i·3-s − 256·4-s + (663. − 1.22e3i)5-s + 1.29e3·6-s + 3.84e3i·7-s + 4.09e3i·8-s − 6.56e3·9-s + (−1.96e4 − 1.06e4i)10-s + 3.67e4·11-s − 2.07e4i·12-s − 1.73e5i·13-s + 6.15e4·14-s + (9.96e4 + 5.37e4i)15-s + 6.55e4·16-s − 4.42e5i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.474 − 0.880i)5-s + 0.408·6-s + 0.605i·7-s + 0.353i·8-s − 0.333·9-s + (−0.622 − 0.335i)10-s + 0.757·11-s − 0.288i·12-s − 1.68i·13-s + 0.428·14-s + (0.508 + 0.274i)15-s + 0.250·16-s − 1.28i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.474 + 0.880i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30\)    =    \(2 \cdot 3 \cdot 5\)
Sign: $-0.474 + 0.880i$
Analytic conductor: \(15.4510\)
Root analytic conductor: \(3.93078\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{30} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 30,\ (\ :9/2),\ -0.474 + 0.880i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.779562 - 1.30643i\)
\(L(\frac12)\) \(\approx\) \(0.779562 - 1.30643i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 16iT \)
3 \( 1 - 81iT \)
5 \( 1 + (-663. + 1.22e3i)T \)
good7 \( 1 - 3.84e3iT - 4.03e7T^{2} \)
11 \( 1 - 3.67e4T + 2.35e9T^{2} \)
13 \( 1 + 1.73e5iT - 1.06e10T^{2} \)
17 \( 1 + 4.42e5iT - 1.18e11T^{2} \)
19 \( 1 + 9.05e5T + 3.22e11T^{2} \)
23 \( 1 + 1.03e6iT - 1.80e12T^{2} \)
29 \( 1 + 5.87e5T + 1.45e13T^{2} \)
31 \( 1 - 6.53e6T + 2.64e13T^{2} \)
37 \( 1 + 2.05e7iT - 1.29e14T^{2} \)
41 \( 1 + 2.35e5T + 3.27e14T^{2} \)
43 \( 1 - 1.82e7iT - 5.02e14T^{2} \)
47 \( 1 - 2.25e7iT - 1.11e15T^{2} \)
53 \( 1 - 6.07e7iT - 3.29e15T^{2} \)
59 \( 1 - 4.24e6T + 8.66e15T^{2} \)
61 \( 1 - 1.40e8T + 1.16e16T^{2} \)
67 \( 1 + 1.49e7iT - 2.72e16T^{2} \)
71 \( 1 + 2.58e8T + 4.58e16T^{2} \)
73 \( 1 + 9.25e7iT - 5.88e16T^{2} \)
79 \( 1 + 8.10e7T + 1.19e17T^{2} \)
83 \( 1 - 1.34e8iT - 1.86e17T^{2} \)
89 \( 1 + 6.43e8T + 3.50e17T^{2} \)
97 \( 1 - 8.53e8iT - 7.60e17T^{2} \)
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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

−14.47912014611600854705520899169, −13.03831271020023785087432640297, −12.10306084723773555326214837769, −10.61831955740693259030205343053, −9.411309827485631610250202531340, −8.432027621614811104581160526174, −5.77754561703288781247709361682, −4.46573282885644780523983726135, −2.57676464804355451924403463149, −0.61924636982346001892719831187, 1.73949056868910590486456529950, 4.02993553056557480900196182953, 6.30281599083837268785189396593, 6.92444252983471329588577885282, 8.585031249883431991362191274425, 10.13567880680423367130812363533, 11.61297754696828790816360178954, 13.29514227057804353247810413111, 14.17945219094058191322894570099, 15.08056505559682266743553661722

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