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Properties

Label 2-1050-15.14-c2-0-48
Degree $2$
Conductor $1050$
Sign $-0.978 - 0.204i$
Analytic cond. $28.6104$
Root an. cond. $5.34887$
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 + (−2.35 − 1.86i)3-s + 2.00·4-s + (3.32 + 2.63i)6-s − 2.64i·7-s − 2.82·8-s + (2.07 + 8.75i)9-s + 19.0i·11-s + (−4.70 − 3.72i)12-s − 1.55i·13-s + 3.74i·14-s + 4.00·16-s + 28.9·17-s + (−2.93 − 12.3i)18-s − 23.3·19-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.784 − 0.620i)3-s + 0.500·4-s + (0.554 + 0.438i)6-s − 0.377i·7-s − 0.353·8-s + (0.230 + 0.973i)9-s + 1.73i·11-s + (−0.392 − 0.310i)12-s − 0.119i·13-s + 0.267i·14-s + 0.250·16-s + 1.70·17-s + (−0.162 − 0.688i)18-s − 1.22·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.204i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.978 - 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.978 - 0.204i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ -0.978 - 0.204i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.01909533909\)
\(L(\frac12)\) \(\approx\) \(0.01909533909\)
\(L(2)\) 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 \( 1 + (2.35 + 1.86i)T \)
5 \( 1 \)
7 \( 1 + 2.64iT \)
good11 \( 1 - 19.0iT - 121T^{2} \)
13 \( 1 + 1.55iT - 169T^{2} \)
17 \( 1 - 28.9T + 289T^{2} \)
19 \( 1 + 23.3T + 361T^{2} \)
23 \( 1 + 23.7T + 529T^{2} \)
29 \( 1 + 18.0iT - 841T^{2} \)
31 \( 1 - 7.88T + 961T^{2} \)
37 \( 1 - 20.5iT - 1.36e3T^{2} \)
41 \( 1 + 50.8iT - 1.68e3T^{2} \)
43 \( 1 + 31.2iT - 1.84e3T^{2} \)
47 \( 1 + 15.7T + 2.20e3T^{2} \)
53 \( 1 + 78.7T + 2.80e3T^{2} \)
59 \( 1 - 89.1iT - 3.48e3T^{2} \)
61 \( 1 - 56.4T + 3.72e3T^{2} \)
67 \( 1 + 56.2iT - 4.48e3T^{2} \)
71 \( 1 - 28.4iT - 5.04e3T^{2} \)
73 \( 1 + 120. iT - 5.32e3T^{2} \)
79 \( 1 - 72.1T + 6.24e3T^{2} \)
83 \( 1 + 104.T + 6.88e3T^{2} \)
89 \( 1 - 36.3iT - 7.92e3T^{2} \)
97 \( 1 - 57.1iT - 9.40e3T^{2} \)
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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

−9.498462918036030887209580662163, −8.147346436784326388811246509420, −7.64058471648509129914968531684, −6.88011205165825183846166014269, −6.08497396622395696826221080269, −5.07004135123443720636496387818, −4.01820657343143786184319051554, −2.32092138915471489252834368423, −1.41269243267209011465092556076, −0.009234704164677922386128262531, 1.24271317127888669153121652113, 2.97504655860273557128566721495, 3.88398031444614476203663465183, 5.24141495927823223761209630400, 5.98900605173203273564443028938, 6.55912690175472649613281759966, 7.977697096546350169790748053726, 8.497182490161091679848678298832, 9.497220164330381500404869003381, 10.09842384953622252916582162474

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