Properties

Label 2-1050-7.6-c2-0-22
Degree $2$
Conductor $1050$
Sign $0.749 - 0.661i$
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 + (4.63 + 5.24i)7-s + 2.82·8-s − 2.99·9-s + 11.7·11-s − 3.46i·12-s + 24.8i·13-s + (6.54 + 7.42i)14-s + 4.00·16-s − 7.26i·17-s − 4.24·18-s + 23.0i·19-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577i·3-s + 0.500·4-s − 0.408i·6-s + (0.661 + 0.749i)7-s + 0.353·8-s − 0.333·9-s + 1.06·11-s − 0.288i·12-s + 1.90i·13-s + (0.467 + 0.530i)14-s + 0.250·16-s − 0.427i·17-s − 0.235·18-s + 1.21i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.749 - 0.661i$
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: \((2,\ 1050,\ (\ :1),\ 0.749 - 0.661i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.204070736\)
\(L(\frac12)\) \(\approx\) \(3.204070736\)
\(L(2)\) 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 \( 1 + (-4.63 - 5.24i)T \)
good11 \( 1 - 11.7T + 121T^{2} \)
13 \( 1 - 24.8iT - 169T^{2} \)
17 \( 1 + 7.26iT - 289T^{2} \)
19 \( 1 - 23.0iT - 361T^{2} \)
23 \( 1 + 26.4T + 529T^{2} \)
29 \( 1 + 57.0T + 841T^{2} \)
31 \( 1 - 10.2iT - 961T^{2} \)
37 \( 1 - 14.2T + 1.36e3T^{2} \)
41 \( 1 - 16.2iT - 1.68e3T^{2} \)
43 \( 1 - 82.3T + 1.84e3T^{2} \)
47 \( 1 - 51.6iT - 2.20e3T^{2} \)
53 \( 1 - 64.8T + 2.80e3T^{2} \)
59 \( 1 + 81.7iT - 3.48e3T^{2} \)
61 \( 1 + 13.1iT - 3.72e3T^{2} \)
67 \( 1 - 22.4T + 4.48e3T^{2} \)
71 \( 1 - 91.5T + 5.04e3T^{2} \)
73 \( 1 + 71.9iT - 5.32e3T^{2} \)
79 \( 1 + 45.2T + 6.24e3T^{2} \)
83 \( 1 - 17.7iT - 6.88e3T^{2} \)
89 \( 1 + 77.5iT - 7.92e3T^{2} \)
97 \( 1 - 6.15iT - 9.40e3T^{2} \)
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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

−9.593946009965993533512256452543, −9.044796212920003988792352714597, −8.000702566195762859314000070038, −7.19492067138098299148218896160, −6.26709198026135341550150160707, −5.70075146973881700860938747154, −4.46483638047201134340311729367, −3.74428871846536144642825723302, −2.17278193079190356343473892131, −1.58605463298922192550122003364, 0.791681657583675627386292827360, 2.34622777897253617117993367122, 3.69575999901942498263753357003, 4.14653141435722381970058219033, 5.30632820107104536343824110402, 5.90971192011260392545212443548, 7.14196632249078204555738142063, 7.83157038408793448203868450583, 8.789107310027205624173006574648, 9.804508361649910385140647759033

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