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Properties

Label 2-1050-105.59-c1-0-11
Degree $2$
Conductor $1050$
Sign $0.00342 - 0.999i$
Analytic cond. $8.38429$
Root an. cond. $2.89556$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.866 + 1.5i)3-s + (−0.499 + 0.866i)4-s + 1.73·6-s + (0.866 + 2.5i)7-s + 0.999·8-s + (−1.5 − 2.59i)9-s + (5.19 + 3i)11-s + (−0.866 − 1.49i)12-s − 1.73·13-s + (1.73 − 2i)14-s + (−0.5 − 0.866i)16-s + (−1.5 + 2.59i)18-s + (1.5 − 0.866i)19-s + (−4.5 − 0.866i)21-s − 6i·22-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.499 + 0.866i)3-s + (−0.249 + 0.433i)4-s + 0.707·6-s + (0.327 + 0.944i)7-s + 0.353·8-s + (−0.5 − 0.866i)9-s + (1.56 + 0.904i)11-s + (−0.250 − 0.433i)12-s − 0.480·13-s + (0.462 − 0.534i)14-s + (−0.125 − 0.216i)16-s + (−0.353 + 0.612i)18-s + (0.344 − 0.198i)19-s + (−0.981 − 0.188i)21-s − 1.27i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00342 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00342 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.00342 - 0.999i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.00342 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.038441625\)
\(L(\frac12)\) \(\approx\) \(1.038441625\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (0.866 - 1.5i)T \)
5 \( 1 \)
7 \( 1 + (-0.866 - 2.5i)T \)
good11 \( 1 + (-5.19 - 3i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.73T + 13T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.5 + 0.866i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 + (3 + 1.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.06 - 3.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + (9 - 5.19i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.19 - 9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.5 - 2.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.52 - 5.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 + (-0.866 + 1.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.3iT - 83T^{2} \)
89 \( 1 + (-5.19 - 9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 15.5T + 97T^{2} \)
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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

−9.893232394665916845091933363061, −9.408180533525510186433979286764, −8.966032397772508467031330275170, −7.77402519599648092276301742038, −6.69759216791007297520150826152, −5.69315671022220113189985602189, −4.74578151945905893176970610984, −3.97480465243556634035248815015, −2.81695578373021165751490618455, −1.45718942623669486517800876856, 0.62563661543478226748887492815, 1.65871806387811295632742885332, 3.47069153867988944583461931377, 4.69104354268968871053500837952, 5.61475415208872578752253924170, 6.70454106121831505465112434962, 6.93581489965783801289761276073, 7.946268946032402637685147756489, 8.692571268675074110387367382052, 9.537376567376632004750358664235

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