Properties

Label 2-1050-7.6-c2-0-39
Degree $2$
Conductor $1050$
Sign $-0.992 - 0.121i$
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 + (−0.853 + 6.94i)7-s − 2.82·8-s − 2.99·9-s + 2.88·11-s − 3.46i·12-s − 13.8i·13-s + (1.20 − 9.82i)14-s + 4.00·16-s + 24.1i·17-s + 4.24·18-s − 6.53i·19-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577i·3-s + 0.500·4-s + 0.408i·6-s + (−0.121 + 0.992i)7-s − 0.353·8-s − 0.333·9-s + 0.261·11-s − 0.288i·12-s − 1.06i·13-s + (0.0861 − 0.701i)14-s + 0.250·16-s + 1.42i·17-s + 0.235·18-s − 0.343i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.121i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.992 - 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.992 - 0.121i$
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: \((2,\ 1050,\ (\ :1),\ -0.992 - 0.121i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.06219265009\)
\(L(\frac12)\) \(\approx\) \(0.06219265009\)
\(L(2)\) 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 \( 1 + (0.853 - 6.94i)T \)
good11 \( 1 - 2.88T + 121T^{2} \)
13 \( 1 + 13.8iT - 169T^{2} \)
17 \( 1 - 24.1iT - 289T^{2} \)
19 \( 1 + 6.53iT - 361T^{2} \)
23 \( 1 + 28.8T + 529T^{2} \)
29 \( 1 - 32.9T + 841T^{2} \)
31 \( 1 + 2.43iT - 961T^{2} \)
37 \( 1 + 50.9T + 1.36e3T^{2} \)
41 \( 1 + 21.5iT - 1.68e3T^{2} \)
43 \( 1 + 13.5T + 1.84e3T^{2} \)
47 \( 1 + 40.7iT - 2.20e3T^{2} \)
53 \( 1 - 17.2T + 2.80e3T^{2} \)
59 \( 1 - 1.47iT - 3.48e3T^{2} \)
61 \( 1 + 111. iT - 3.72e3T^{2} \)
67 \( 1 + 120.T + 4.48e3T^{2} \)
71 \( 1 + 90.3T + 5.04e3T^{2} \)
73 \( 1 - 21.4iT - 5.32e3T^{2} \)
79 \( 1 - 66.1T + 6.24e3T^{2} \)
83 \( 1 - 78.5iT - 6.88e3T^{2} \)
89 \( 1 - 90.9iT - 7.92e3T^{2} \)
97 \( 1 + 44.1iT - 9.40e3T^{2} \)
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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

−9.111617404566418764203388197412, −8.370314627608094957266514820306, −7.931514941288247765245000033738, −6.73150053990607678148570910137, −6.07086128768614752957107299013, −5.25532081635644172602901003876, −3.65085139959642129282640039341, −2.53810416408907629396355850991, −1.55068958920343672766935492373, −0.02436939212777101188929617006, 1.42650306753293439888644360054, 2.86593908604901864543173488934, 4.00126976685593565538081729647, 4.78644694695130524402931260379, 6.10783962869203855712609823630, 6.94314085782451910912803853649, 7.65833481848891307093292186757, 8.687242682159363597856495345997, 9.386722274455433754666880445885, 10.13184331326161985727743673335

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