Properties

Label 2-1050-7.5-c2-0-31
Degree $2$
Conductor $1050$
Sign $0.744 + 0.667i$
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  + (−0.707 + 1.22i)2-s + (−1.5 + 0.866i)3-s + (−0.999 − 1.73i)4-s − 2.44i·6-s + (3.97 − 5.76i)7-s + 2.82·8-s + (1.5 − 2.59i)9-s + (0.647 + 1.12i)11-s + (2.99 + 1.73i)12-s − 3.22i·13-s + (4.24 + 8.94i)14-s + (−2.00 + 3.46i)16-s + (−26.2 + 15.1i)17-s + (2.12 + 3.67i)18-s + (28.9 + 16.7i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.5 + 0.288i)3-s + (−0.249 − 0.433i)4-s − 0.408i·6-s + (0.567 − 0.823i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (0.0588 + 0.101i)11-s + (0.249 + 0.144i)12-s − 0.248i·13-s + (0.303 + 0.638i)14-s + (−0.125 + 0.216i)16-s + (−1.54 + 0.893i)17-s + (0.117 + 0.204i)18-s + (1.52 + 0.879i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.744 + 0.667i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ 0.744 + 0.667i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.007949299\)
\(L(\frac12)\) \(\approx\) \(1.007949299\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 1.22i)T \)
3 \( 1 + (1.5 - 0.866i)T \)
5 \( 1 \)
7 \( 1 + (-3.97 + 5.76i)T \)
good11 \( 1 + (-0.647 - 1.12i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 3.22iT - 169T^{2} \)
17 \( 1 + (26.2 - 15.1i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-28.9 - 16.7i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-17.8 + 30.9i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 32.8T + 841T^{2} \)
31 \( 1 + (-39.5 + 22.8i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (20.0 - 34.6i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 42.2iT - 1.68e3T^{2} \)
43 \( 1 + 55.4T + 1.84e3T^{2} \)
47 \( 1 + (52.6 + 30.3i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (27.3 + 47.3i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-11.4 + 6.59i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-34.1 - 19.7i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-20.0 - 34.7i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 46.4T + 5.04e3T^{2} \)
73 \( 1 + (-118. + 68.2i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-21.1 + 36.6i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 1.53iT - 6.88e3T^{2} \)
89 \( 1 + (30.2 + 17.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 150. iT - 9.40e3T^{2} \)
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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

−9.763575020567742468363224757873, −8.617985769874011016254337696973, −8.022266287087022805131650643197, −6.99106488183580217621906957928, −6.41418210652350469418045895178, −5.27962618663968780364972988575, −4.58053615835075713879382504259, −3.59172109975424691215827103543, −1.75839932381316687749551591279, −0.43686286437836542698765397393, 1.11497410341314863390173436378, 2.24831264125018986694286803552, 3.28209484416867968944455627230, 4.81069711709561102180813378762, 5.24481748373847797802190106213, 6.57162749256153374046428280656, 7.34340905783598724582982893304, 8.269169679854072387560454957475, 9.258985346390398177155636397588, 9.559954193238489675074706821604

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