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Properties

Label 2-30-15.14-c6-0-1
Degree $2$
Conductor $30$
Sign $-0.436 - 0.899i$
Analytic cond. $6.90162$
Root an. cond. $2.62709$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.65·2-s + (0.305 + 26.9i)3-s + 32.0·4-s + (113. − 53.3i)5-s + (−1.72 − 152. i)6-s + 274. i·7-s − 181.·8-s + (−728. + 16.4i)9-s + (−639. + 301. i)10-s + 684. i·11-s + (9.77 + 863. i)12-s + 3.45e3i·13-s − 1.55e3i·14-s + (1.47e3 + 3.03e3i)15-s + 1.02e3·16-s − 8.13e3·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.0113 + 0.999i)3-s + 0.500·4-s + (0.904 − 0.426i)5-s + (−0.00800 − 0.707i)6-s + 0.799i·7-s − 0.353·8-s + (−0.999 + 0.0226i)9-s + (−0.639 + 0.301i)10-s + 0.513i·11-s + (0.00565 + 0.499i)12-s + 1.57i·13-s − 0.565i·14-s + (0.436 + 0.899i)15-s + 0.250·16-s − 1.65·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.436 - 0.899i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.436 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30\)    =    \(2 \cdot 3 \cdot 5\)
Sign: $-0.436 - 0.899i$
Analytic conductor: \(6.90162\)
Root analytic conductor: \(2.62709\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{30} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 30,\ (\ :3),\ -0.436 - 0.899i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.592502 + 0.946340i\)
\(L(\frac12)\) \(\approx\) \(0.592502 + 0.946340i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 5.65T \)
3 \( 1 + (-0.305 - 26.9i)T \)
5 \( 1 + (-113. + 53.3i)T \)
good7 \( 1 - 274. iT - 1.17e5T^{2} \)
11 \( 1 - 684. iT - 1.77e6T^{2} \)
13 \( 1 - 3.45e3iT - 4.82e6T^{2} \)
17 \( 1 + 8.13e3T + 2.41e7T^{2} \)
19 \( 1 - 1.40e3T + 4.70e7T^{2} \)
23 \( 1 - 3.94e3T + 1.48e8T^{2} \)
29 \( 1 - 3.09e4iT - 5.94e8T^{2} \)
31 \( 1 - 3.53e4T + 8.87e8T^{2} \)
37 \( 1 - 6.49e3iT - 2.56e9T^{2} \)
41 \( 1 + 8.91e4iT - 4.75e9T^{2} \)
43 \( 1 - 9.79e4iT - 6.32e9T^{2} \)
47 \( 1 - 1.02e5T + 1.07e10T^{2} \)
53 \( 1 - 3.48e4T + 2.21e10T^{2} \)
59 \( 1 + 1.90e4iT - 4.21e10T^{2} \)
61 \( 1 + 8.33e4T + 5.15e10T^{2} \)
67 \( 1 + 3.42e5iT - 9.04e10T^{2} \)
71 \( 1 + 6.24e5iT - 1.28e11T^{2} \)
73 \( 1 + 3.44e5iT - 1.51e11T^{2} \)
79 \( 1 - 2.31e5T + 2.43e11T^{2} \)
83 \( 1 - 1.41e5T + 3.26e11T^{2} \)
89 \( 1 - 1.12e6iT - 4.96e11T^{2} \)
97 \( 1 + 5.05e5iT - 8.32e11T^{2} \)
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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

−16.22095673904610937206283102049, −15.19199135160860996248628397987, −13.85046186738193037233271173377, −12.06354094831242356184754431119, −10.73732090383454050851604607469, −9.310392470864069490007127153567, −8.893802738534445228394816193377, −6.43480560108707412802978720814, −4.76926790037118841766834969380, −2.20606846782017711469511698840, 0.75637813443508835336874461009, 2.61227752860129505119375445670, 5.96715068768754529635811405592, 7.17733142053949783419905035596, 8.528904253554166252082502671902, 10.20038538491875679186285245031, 11.26835516073118984588798813333, 13.05724514039814243607912371517, 13.80736053899975355911730593071, 15.35289636971997722397439583599

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