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Properties

Label 2-30-15.14-c2-0-3
Degree $2$
Conductor $30$
Sign $0.934 + 0.355i$
Analytic cond. $0.817440$
Root an. cond. $0.904124$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s + (−0.707 − 2.91i)3-s + 2.00·4-s + (−2.82 + 4.12i)5-s + (−1.00 − 4.12i)6-s + 5.83i·7-s + 2.82·8-s + (−8 + 4.12i)9-s + (−4.00 + 5.83i)10-s − 16.4i·11-s + (−1.41 − 5.83i)12-s + 8.24i·14-s + (14.0 + 5.33i)15-s + 4.00·16-s − 11.3·17-s + (−11.3 + 5.83i)18-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.235 − 0.971i)3-s + 0.500·4-s + (−0.565 + 0.824i)5-s + (−0.166 − 0.687i)6-s + 0.832i·7-s + 0.353·8-s + (−0.888 + 0.458i)9-s + (−0.400 + 0.583i)10-s − 1.49i·11-s + (−0.117 − 0.485i)12-s + 0.589i·14-s + (0.934 + 0.355i)15-s + 0.250·16-s − 0.665·17-s + (−0.628 + 0.323i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30\)    =    \(2 \cdot 3 \cdot 5\)
Sign: $0.934 + 0.355i$
Analytic conductor: \(0.817440\)
Root analytic conductor: \(0.904124\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{30} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 30,\ (\ :1),\ 0.934 + 0.355i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.19370 - 0.219267i\)
\(L(\frac12)\) \(\approx\) \(1.19370 - 0.219267i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 + (0.707 + 2.91i)T \)
5 \( 1 + (2.82 - 4.12i)T \)
good7 \( 1 - 5.83iT - 49T^{2} \)
11 \( 1 + 16.4iT - 121T^{2} \)
13 \( 1 - 169T^{2} \)
17 \( 1 + 11.3T + 289T^{2} \)
19 \( 1 - 12T + 361T^{2} \)
23 \( 1 - 24.0T + 529T^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 + 32T + 961T^{2} \)
37 \( 1 - 23.3iT - 1.36e3T^{2} \)
41 \( 1 - 57.7iT - 1.68e3T^{2} \)
43 \( 1 + 40.8iT - 1.84e3T^{2} \)
47 \( 1 + 35.3T + 2.20e3T^{2} \)
53 \( 1 - 67.8T + 2.80e3T^{2} \)
59 \( 1 + 16.4iT - 3.48e3T^{2} \)
61 \( 1 + 16T + 3.72e3T^{2} \)
67 \( 1 + 5.83iT - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 116. iT - 5.32e3T^{2} \)
79 \( 1 + 72T + 6.24e3T^{2} \)
83 \( 1 + 43.8T + 6.88e3T^{2} \)
89 \( 1 - 65.9iT - 7.92e3T^{2} \)
97 \( 1 + 163. iT - 9.40e3T^{2} \)
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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

−16.58882178716826748786556648566, −15.33053636789295574531251394776, −14.16663333127744875340328517364, −13.08710303284183496334975798865, −11.71983661285254769934238067595, −11.06588862538936461485155837294, −8.466402456698073833816059007423, −6.95497593315290938568200687329, −5.69296378855426107335447294381, −2.97760111814052126250815104749, 4.01457074796258314465551376197, 5.06168367233999760498139805777, 7.28251789770048511945899769940, 9.238455499738404659373897359660, 10.67299930059362654735356988374, 11.93040687964868699036428154939, 13.12872899599409982934838582189, 14.67942219432902335714473151718, 15.61919684673188299507500018522, 16.60555277562872714669791982688

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