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Properties

Label 2-1050-5.4-c5-0-53
Degree $2$
Conductor $1050$
Sign $0.894 + 0.447i$
Analytic cond. $168.403$
Root an. cond. $12.9770$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4i·2-s + 9i·3-s − 16·4-s + 36·6-s + 49i·7-s + 64i·8-s − 81·9-s + 216·11-s − 144i·12-s − 998i·13-s + 196·14-s + 256·16-s + 1.30e3i·17-s + 324i·18-s − 884·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.377i·7-s + 0.353i·8-s − 0.333·9-s + 0.538·11-s − 0.288i·12-s − 1.63i·13-s + 0.267·14-s + 0.250·16-s + 1.09i·17-s + 0.235i·18-s − 0.561·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(168.403\)
Root analytic conductor: \(12.9770\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (799, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :5/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.877771993\)
\(L(\frac12)\) \(\approx\) \(1.877771993\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 4iT \)
3 \( 1 - 9iT \)
5 \( 1 \)
7 \( 1 - 49iT \)
good11 \( 1 - 216T + 1.61e5T^{2} \)
13 \( 1 + 998iT - 3.71e5T^{2} \)
17 \( 1 - 1.30e3iT - 1.41e6T^{2} \)
19 \( 1 + 884T + 2.47e6T^{2} \)
23 \( 1 - 2.26e3iT - 6.43e6T^{2} \)
29 \( 1 - 1.48e3T + 2.05e7T^{2} \)
31 \( 1 - 8.36e3T + 2.86e7T^{2} \)
37 \( 1 + 4.71e3iT - 6.93e7T^{2} \)
41 \( 1 + 9.78e3T + 1.15e8T^{2} \)
43 \( 1 + 1.94e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.22e4iT - 2.29e8T^{2} \)
53 \( 1 + 2.67e4iT - 4.18e8T^{2} \)
59 \( 1 + 2.80e4T + 7.14e8T^{2} \)
61 \( 1 + 3.88e4T + 8.44e8T^{2} \)
67 \( 1 - 2.39e4iT - 1.35e9T^{2} \)
71 \( 1 + 2.06e4T + 1.80e9T^{2} \)
73 \( 1 + 290iT - 2.07e9T^{2} \)
79 \( 1 - 9.95e4T + 3.07e9T^{2} \)
83 \( 1 + 1.93e4iT - 3.93e9T^{2} \)
89 \( 1 + 3.63e4T + 5.58e9T^{2} \)
97 \( 1 + 7.90e4iT - 8.58e9T^{2} \)
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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

−9.211433126041990958408015172270, −8.468916438998643056892091228891, −7.78536465460013522213364755446, −6.33368499683026729038060166761, −5.58248338769560090595730521320, −4.65644010017826025905763986868, −3.66516338241220223454728099492, −2.94747374557849682972651990727, −1.76938231399989872068125491488, −0.57386559701590160890638792612, 0.62661402488640268972036140659, 1.69944604946496315646301623660, 2.94751200962044276835549287307, 4.30734053220898864157231200143, 4.82667214081940254692594484588, 6.37777010966495410606434202864, 6.54812730040931987597674076735, 7.43964068391740601324538375369, 8.346897353237490953036070411316, 9.075656973880108183425007232434

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