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Properties

Label 2-1050-7.6-c2-0-44
Degree $2$
Conductor $1050$
Sign $0.499 + 0.866i$
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 + 1.73i·3-s + 2.00·4-s + 2.44i·6-s + (6.06 − 3.49i)7-s + 2.82·8-s − 2.99·9-s − 17.5·11-s + 3.46i·12-s − 17.6i·13-s + (8.57 − 4.94i)14-s + 4.00·16-s − 14.0i·17-s − 4.24·18-s − 16.6i·19-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577i·3-s + 0.500·4-s + 0.408i·6-s + (0.866 − 0.499i)7-s + 0.353·8-s − 0.333·9-s − 1.59·11-s + 0.288i·12-s − 1.35i·13-s + (0.612 − 0.353i)14-s + 0.250·16-s − 0.828i·17-s − 0.235·18-s − 0.873i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.499 + 0.866i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ 0.499 + 0.866i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.620166902\)
\(L(\frac12)\) \(\approx\) \(2.620166902\)
\(L(2)\) 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 \( 1 - 1.73iT \)
5 \( 1 \)
7 \( 1 + (-6.06 + 3.49i)T \)
good11 \( 1 + 17.5T + 121T^{2} \)
13 \( 1 + 17.6iT - 169T^{2} \)
17 \( 1 + 14.0iT - 289T^{2} \)
19 \( 1 + 16.6iT - 361T^{2} \)
23 \( 1 - 15.5T + 529T^{2} \)
29 \( 1 - 36.9T + 841T^{2} \)
31 \( 1 + 54.2iT - 961T^{2} \)
37 \( 1 + 45.0T + 1.36e3T^{2} \)
41 \( 1 + 1.20iT - 1.68e3T^{2} \)
43 \( 1 + 50.7T + 1.84e3T^{2} \)
47 \( 1 - 91.7iT - 2.20e3T^{2} \)
53 \( 1 - 51.5T + 2.80e3T^{2} \)
59 \( 1 + 14.8iT - 3.48e3T^{2} \)
61 \( 1 - 91.2iT - 3.72e3T^{2} \)
67 \( 1 - 5.61T + 4.48e3T^{2} \)
71 \( 1 - 80.8T + 5.04e3T^{2} \)
73 \( 1 + 91.0iT - 5.32e3T^{2} \)
79 \( 1 + 28.5T + 6.24e3T^{2} \)
83 \( 1 + 50.3iT - 6.88e3T^{2} \)
89 \( 1 + 76.5iT - 7.92e3T^{2} \)
97 \( 1 - 125. iT - 9.40e3T^{2} \)
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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

−9.889152849897460001098899374959, −8.594689809638483547071550101527, −7.82854332983057777674602818311, −7.18121021175419729512003476110, −5.81425769699409443837970629789, −5.02435008989515469862027003964, −4.60538557884880869160627392163, −3.21135424635528015858546054333, −2.48871138579965087064377021140, −0.61418027144234117955949671007, 1.56607514538672291683405740663, 2.37296884584890061149721840786, 3.58024494961281824245192489329, 4.91746745419131970533290490838, 5.33592830381557214271169737395, 6.49842278746405466044863074851, 7.18959618421282950421700459438, 8.307309541914880115395857500925, 8.588038693122857740454340933284, 10.14052006588910267815152391608

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