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Properties

Label 2-1050-7.5-c2-0-23
Degree $2$
Conductor $1050$
Sign $0.586 + 0.809i$
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 + (5.10 − 4.79i)7-s − 2.82·8-s + (1.5 − 2.59i)9-s + (9.03 + 15.6i)11-s + (2.99 + 1.73i)12-s − 18.6i·13-s + (−2.26 − 9.63i)14-s + (−2.00 + 3.46i)16-s + (−1.33 + 0.770i)17-s + (−2.12 − 3.67i)18-s + (29.4 + 17.0i)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 + (0.728 − 0.684i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (0.821 + 1.42i)11-s + (0.249 + 0.144i)12-s − 1.43i·13-s + (−0.161 − 0.688i)14-s + (−0.125 + 0.216i)16-s + (−0.0785 + 0.0453i)17-s + (−0.117 − 0.204i)18-s + (1.55 + 0.895i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 + 0.809i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.586 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.586 + 0.809i$
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: \((2,\ 1050,\ (\ :1),\ 0.586 + 0.809i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.195374085\)
\(L(\frac12)\) \(\approx\) \(2.195374085\)
\(L(2)\) 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 \( 1 + (-5.10 + 4.79i)T \)
good11 \( 1 + (-9.03 - 15.6i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 18.6iT - 169T^{2} \)
17 \( 1 + (1.33 - 0.770i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-29.4 - 17.0i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (13.4 - 23.3i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 16.4T + 841T^{2} \)
31 \( 1 + (24.1 - 13.9i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-25.8 + 44.7i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 37.4iT - 1.68e3T^{2} \)
43 \( 1 - 63.6T + 1.84e3T^{2} \)
47 \( 1 + (-24.4 - 14.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-0.221 - 0.384i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-64.2 + 37.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-91.0 - 52.5i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (6.35 + 11.0i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 45.7T + 5.04e3T^{2} \)
73 \( 1 + (31.5 - 18.2i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-66.5 + 115. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 49.9iT - 6.88e3T^{2} \)
89 \( 1 + (-85.9 - 49.6i)T + (3.96e3 + 6.85e3i)T^{2} \)
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

−9.877966585126191431854809181830, −9.097673107539796866991984683827, −7.63834975675821145331218935874, −7.30906430200466155193534219290, −5.77700295610364692798434749682, −5.26110742450991210158223325238, −4.16861551772307983947645499620, −3.55652720840844302438485854147, −1.93073744999530870902042553500, −0.882826247671788147748404702660, 0.993048252108429866755373315448, 2.45891135084769890298953386800, 3.84547015094946783370914985985, 4.79612482859221614406373751267, 5.68773670162137218803251548095, 6.35313199268444300512080814694, 7.16823414902760403891037743294, 8.140161849424425499345596042967, 8.918690474238775840520630954475, 9.526926080232866495301598137534

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