Properties

Label 2-1050-15.14-c2-0-53
Degree $2$
Conductor $1050$
Sign $0.0980 + 0.995i$
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 + (−2.53 + 1.59i)3-s + 2.00·4-s + (−3.59 + 2.26i)6-s − 2.64i·7-s + 2.82·8-s + (3.89 − 8.11i)9-s + 7.44i·11-s + (−5.07 + 3.19i)12-s − 13.9i·13-s − 3.74i·14-s + 4.00·16-s + 17.8·17-s + (5.50 − 11.4i)18-s − 33.6·19-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.846 + 0.532i)3-s + 0.500·4-s + (−0.598 + 0.376i)6-s − 0.377i·7-s + 0.353·8-s + (0.432 − 0.901i)9-s + 0.677i·11-s + (−0.423 + 0.266i)12-s − 1.07i·13-s − 0.267i·14-s + 0.250·16-s + 1.05·17-s + (0.305 − 0.637i)18-s − 1.77·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0980 + 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0980 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.0980 + 0.995i$
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: \((2,\ 1050,\ (\ :1),\ 0.0980 + 0.995i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.451941123\)
\(L(\frac12)\) \(\approx\) \(1.451941123\)
\(L(2)\) 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 \( 1 + (2.53 - 1.59i)T \)
5 \( 1 \)
7 \( 1 + 2.64iT \)
good11 \( 1 - 7.44iT - 121T^{2} \)
13 \( 1 + 13.9iT - 169T^{2} \)
17 \( 1 - 17.8T + 289T^{2} \)
19 \( 1 + 33.6T + 361T^{2} \)
23 \( 1 + 7.94T + 529T^{2} \)
29 \( 1 + 8.60iT - 841T^{2} \)
31 \( 1 + 56.7T + 961T^{2} \)
37 \( 1 + 12.0iT - 1.36e3T^{2} \)
41 \( 1 + 75.2iT - 1.68e3T^{2} \)
43 \( 1 + 60.9iT - 1.84e3T^{2} \)
47 \( 1 - 67.5T + 2.20e3T^{2} \)
53 \( 1 - 49.1T + 2.80e3T^{2} \)
59 \( 1 + 82.5iT - 3.48e3T^{2} \)
61 \( 1 + 6.51T + 3.72e3T^{2} \)
67 \( 1 - 27.8iT - 4.48e3T^{2} \)
71 \( 1 + 1.41iT - 5.04e3T^{2} \)
73 \( 1 + 46.2iT - 5.32e3T^{2} \)
79 \( 1 - 49.2T + 6.24e3T^{2} \)
83 \( 1 + 42.4T + 6.88e3T^{2} \)
89 \( 1 - 63.3iT - 7.92e3T^{2} \)
97 \( 1 + 34.3iT - 9.40e3T^{2} \)
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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

−9.860299740417430204069239419678, −8.763713082578128750276795267565, −7.56232227036808988988385089887, −6.88321766499441501482622289011, −5.77241703280239172369778200272, −5.33379717130904595116138013530, −4.17815741576048845052321075991, −3.61516645241098780950361499730, −2.05665398896396758984322172793, −0.39241525725608213944348232940, 1.37924328787969669584809327186, 2.47376675232360784749777851785, 3.87642114703993698462415317868, 4.78665794252246889156334692474, 5.79836947782558303691033124418, 6.27374482687982934115146725246, 7.16609667919166236900732076253, 8.071471314752249439314178032020, 9.032310968234492642198623430468, 10.19185399170679960573133084618

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