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Properties

Label 2-30-5.3-c2-0-1
Degree $2$
Conductor $30$
Sign $0.728 + 0.685i$
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 − i)2-s + (−1.22 − 1.22i)3-s − 2i·4-s + (4.89 + i)5-s − 2.44·6-s + (−8.89 + 8.89i)7-s + (−2 − 2i)8-s + 2.99i·9-s + (5.89 − 3.89i)10-s + 5.79·11-s + (−2.44 + 2.44i)12-s + (−6.79 − 6.79i)13-s + 17.7i·14-s + (−4.77 − 7.22i)15-s − 4·16-s + (6.10 − 6.10i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (−0.408 − 0.408i)3-s − 0.5i·4-s + (0.979 + 0.200i)5-s − 0.408·6-s + (−1.27 + 1.27i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (0.589 − 0.389i)10-s + 0.527·11-s + (−0.204 + 0.204i)12-s + (−0.522 − 0.522i)13-s + 1.27i·14-s + (−0.318 − 0.481i)15-s − 0.250·16-s + (0.358 − 0.358i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.685i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.728 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30\)    =    \(2 \cdot 3 \cdot 5\)
Sign: $0.728 + 0.685i$
Analytic conductor: \(0.817440\)
Root analytic conductor: \(0.904124\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{30} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 30,\ (\ :1),\ 0.728 + 0.685i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.05255 - 0.417314i\)
\(L(\frac12)\) \(\approx\) \(1.05255 - 0.417314i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
3 \( 1 + (1.22 + 1.22i)T \)
5 \( 1 + (-4.89 - i)T \)
good7 \( 1 + (8.89 - 8.89i)T - 49iT^{2} \)
11 \( 1 - 5.79T + 121T^{2} \)
13 \( 1 + (6.79 + 6.79i)T + 169iT^{2} \)
17 \( 1 + (-6.10 + 6.10i)T - 289iT^{2} \)
19 \( 1 + 6.20iT - 361T^{2} \)
23 \( 1 + (18.6 + 18.6i)T + 529iT^{2} \)
29 \( 1 - 6.20iT - 841T^{2} \)
31 \( 1 + 0.404T + 961T^{2} \)
37 \( 1 + (27 - 27i)T - 1.36e3iT^{2} \)
41 \( 1 + 1.79T + 1.68e3T^{2} \)
43 \( 1 + (-36.4 - 36.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (-38.6 + 38.6i)T - 2.20e3iT^{2} \)
53 \( 1 + (-69.0 - 69.0i)T + 2.80e3iT^{2} \)
59 \( 1 + 20iT - 3.48e3T^{2} \)
61 \( 1 + 63.1T + 3.72e3T^{2} \)
67 \( 1 + (-40.0 + 40.0i)T - 4.48e3iT^{2} \)
71 \( 1 - 25.7T + 5.04e3T^{2} \)
73 \( 1 + (56.7 + 56.7i)T + 5.32e3iT^{2} \)
79 \( 1 - 139. iT - 6.24e3T^{2} \)
83 \( 1 + (-13.7 - 13.7i)T + 6.88e3iT^{2} \)
89 \( 1 + 58.6iT - 7.92e3T^{2} \)
97 \( 1 + (15.9 - 15.9i)T - 9.40e3iT^{2} \)
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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

−16.70078131114902956295518357886, −15.32070452111438941116796674831, −13.96084791158615434346475167694, −12.75942140833946660143196755059, −12.03573357772855083898470178735, −10.28518989466324697775697227981, −9.187368065725560403528322552354, −6.54895168565730344583305656571, −5.53373706520711472499091565625, −2.65158771136712994184584443690, 3.95063285925117270929903707532, 5.85391794536876729071588751358, 7.04669331068887754683718107643, 9.394821779975947401099552855393, 10.37146705466737872868219592587, 12.27079893407829307417435633175, 13.46968007082859793744481544312, 14.34269931708502939750933208666, 16.02714855756204109280057578926, 16.82824346406776693310573806257

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