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Properties

Label 2-1050-7.3-c2-0-45
Degree $2$
Conductor $1050$
Sign $0.965 + 0.262i$
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  + (0.707 + 1.22i)2-s + (1.5 + 0.866i)3-s + (−0.999 + 1.73i)4-s + 2.44i·6-s + (2.67 − 6.46i)7-s − 2.82·8-s + (1.5 + 2.59i)9-s + (−0.578 + 1.00i)11-s + (−2.99 + 1.73i)12-s − 14.8i·13-s + (9.81 − 1.29i)14-s + (−2.00 − 3.46i)16-s + (−10.9 − 6.30i)17-s + (−2.12 + 3.67i)18-s + (16.7 − 9.65i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.5 + 0.288i)3-s + (−0.249 + 0.433i)4-s + 0.408i·6-s + (0.381 − 0.924i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (−0.0525 + 0.0910i)11-s + (−0.249 + 0.144i)12-s − 1.13i·13-s + (0.700 − 0.0928i)14-s + (−0.125 − 0.216i)16-s + (−0.642 − 0.371i)17-s + (−0.117 + 0.204i)18-s + (0.879 − 0.507i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.262i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.965 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.965 + 0.262i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ 0.965 + 0.262i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.503577583\)
\(L(\frac12)\) \(\approx\) \(2.503577583\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 1.22i)T \)
3 \( 1 + (-1.5 - 0.866i)T \)
5 \( 1 \)
7 \( 1 + (-2.67 + 6.46i)T \)
good11 \( 1 + (0.578 - 1.00i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 14.8iT - 169T^{2} \)
17 \( 1 + (10.9 + 6.30i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-16.7 + 9.65i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (12.1 + 21.0i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 49.0T + 841T^{2} \)
31 \( 1 + (24.9 + 14.4i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (26.6 + 46.1i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 38.0iT - 1.68e3T^{2} \)
43 \( 1 - 63.5T + 1.84e3T^{2} \)
47 \( 1 + (21.8 - 12.5i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-10.4 + 18.0i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-21.1 - 12.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-5.53 + 3.19i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-62.2 + 107. i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 118.T + 5.04e3T^{2} \)
73 \( 1 + (-34.2 - 19.7i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-46.4 - 80.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 5.79iT - 6.88e3T^{2} \)
89 \( 1 + (-131. + 75.9i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 144. iT - 9.40e3T^{2} \)
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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

−9.617355307701486094205454616252, −8.687980535357130770090758464835, −7.889389263964278086939511982757, −7.31629923111300224272720718306, −6.36788260623123300288558663619, −5.20008697811324806601850127082, −4.50253144122824387910656632772, −3.55568720731984174084021763690, −2.49794064845703055818942910237, −0.66351400924634395042289745504, 1.43589444632637781234057977080, 2.27383338552278358209190451532, 3.33830390246261684855104494736, 4.36556335187956452831626051927, 5.35623192144243700031062760820, 6.26918913235045105419319281067, 7.24816425150538202595755225137, 8.357189461592207170865575605444, 8.937624764720130150628510219435, 9.692889728423625167476432243430

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