Properties

Label 2-1050-35.24-c2-0-23
Degree $2$
Conductor $1050$
Sign $0.847 + 0.530i$
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  + (1.22 − 0.707i)2-s + (0.866 − 1.5i)3-s + (0.999 − 1.73i)4-s − 2.44i·6-s + (2.86 + 6.38i)7-s − 2.82i·8-s + (−1.5 − 2.59i)9-s + (−9.98 + 17.2i)11-s + (−1.73 − 2.99i)12-s − 3.49·13-s + (8.02 + 5.80i)14-s + (−2.00 − 3.46i)16-s + (9.12 − 15.7i)17-s + (−3.67 − 2.12i)18-s + (21.3 − 12.3i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.288 − 0.5i)3-s + (0.249 − 0.433i)4-s − 0.408i·6-s + (0.408 + 0.912i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.907 + 1.57i)11-s + (−0.144 − 0.249i)12-s − 0.269·13-s + (0.572 + 0.414i)14-s + (−0.125 − 0.216i)16-s + (0.536 − 0.929i)17-s + (−0.204 − 0.117i)18-s + (1.12 − 0.647i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.847 + 0.530i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ 0.847 + 0.530i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.253254645\)
\(L(\frac12)\) \(\approx\) \(3.253254645\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
3 \( 1 + (-0.866 + 1.5i)T \)
5 \( 1 \)
7 \( 1 + (-2.86 - 6.38i)T \)
good11 \( 1 + (9.98 - 17.2i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 3.49T + 169T^{2} \)
17 \( 1 + (-9.12 + 15.7i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-21.3 + 12.3i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-21.8 + 12.5i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 53.1T + 841T^{2} \)
31 \( 1 + (-26.0 - 15.0i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-40.5 + 23.3i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 31.5iT - 1.68e3T^{2} \)
43 \( 1 - 64.4iT - 1.84e3T^{2} \)
47 \( 1 + (-14.0 - 24.3i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-56.1 - 32.4i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (86.7 + 50.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-6.94 + 4.01i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-14.0 - 8.13i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 107.T + 5.04e3T^{2} \)
73 \( 1 + (25.8 - 44.7i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (10.9 + 19.0i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 0.417T + 6.88e3T^{2} \)
89 \( 1 + (-96.3 + 55.6i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 74.2T + 9.40e3T^{2} \)
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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

−9.669307847789920915773929222817, −8.933346668843456701509821447231, −7.75958684502507513705053254466, −7.23073234957400720714848909536, −6.18310770278231807001088391287, −4.96028253358974915634332643838, −4.75087731233427712282671655869, −2.80884010804755691674922087045, −2.56607244789170499790834106560, −1.10498836002760038932962353302, 1.00683514346438915311510987100, 2.87050338731856512627792843333, 3.53559290990090050446005333937, 4.57816612288908802059841648749, 5.41532674699707894776741896232, 6.20949787566174774078723011934, 7.44253581690896170292058115300, 8.051644959423957845090919671893, 8.698511635471899766996232107922, 10.05915411568721375202677289320

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