Properties

Label 2-1050-7.6-c2-0-25
Degree $2$
Conductor $1050$
Sign $0.974 - 0.225i$
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 + (1.57 + 6.81i)7-s − 2.82·8-s − 2.99·9-s − 8.15·11-s + 3.46i·12-s − 14.6i·13-s + (−2.23 − 9.64i)14-s + 4.00·16-s + 5.81i·17-s + 4.24·18-s − 33.7i·19-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577i·3-s + 0.500·4-s − 0.408i·6-s + (0.225 + 0.974i)7-s − 0.353·8-s − 0.333·9-s − 0.741·11-s + 0.288i·12-s − 1.12i·13-s + (−0.159 − 0.688i)14-s + 0.250·16-s + 0.342i·17-s + 0.235·18-s − 1.77i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.974 - 0.225i$
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: \((2,\ 1050,\ (\ :1),\ 0.974 - 0.225i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.248738644\)
\(L(\frac12)\) \(\approx\) \(1.248738644\)
\(L(2)\) 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 \( 1 + (-1.57 - 6.81i)T \)
good11 \( 1 + 8.15T + 121T^{2} \)
13 \( 1 + 14.6iT - 169T^{2} \)
17 \( 1 - 5.81iT - 289T^{2} \)
19 \( 1 + 33.7iT - 361T^{2} \)
23 \( 1 - 37.2T + 529T^{2} \)
29 \( 1 - 9.25T + 841T^{2} \)
31 \( 1 + 19.2iT - 961T^{2} \)
37 \( 1 - 63.4T + 1.36e3T^{2} \)
41 \( 1 - 8.25iT - 1.68e3T^{2} \)
43 \( 1 + 42.0T + 1.84e3T^{2} \)
47 \( 1 - 23.3iT - 2.20e3T^{2} \)
53 \( 1 - 71.3T + 2.80e3T^{2} \)
59 \( 1 - 42.9iT - 3.48e3T^{2} \)
61 \( 1 + 34.2iT - 3.72e3T^{2} \)
67 \( 1 - 4.99T + 4.48e3T^{2} \)
71 \( 1 + 38.8T + 5.04e3T^{2} \)
73 \( 1 + 124. iT - 5.32e3T^{2} \)
79 \( 1 - 56.1T + 6.24e3T^{2} \)
83 \( 1 - 90.3iT - 6.88e3T^{2} \)
89 \( 1 + 16.2iT - 7.92e3T^{2} \)
97 \( 1 - 82.7iT - 9.40e3T^{2} \)
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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

−9.611211076641843680958109736487, −8.985883953199646595096216504227, −8.272862072391690467815373159310, −7.48481922677759637152537749980, −6.34943167219175801158298434607, −5.39922650020626059921536454143, −4.72813818611067961592157135694, −3.06543415914886921764881730322, −2.48183225547135427162226815414, −0.67656263667089504789955062196, 0.868863768381750448710895648200, 1.89131506436671914807147648220, 3.17652825376759235271583297887, 4.39741494714458217752159462065, 5.54721896378215602601832475819, 6.67361203338321319577779388980, 7.23724268212017451050999550083, 8.002668944848348010457321476937, 8.730909155024972759037991269890, 9.758541493860087183504567979195

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