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Properties

Label 2-1050-7.3-c2-0-26
Degree $2$
Conductor $1050$
Sign $0.989 + 0.145i$
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 + (1.88 − 6.74i)7-s + 2.82·8-s + (1.5 + 2.59i)9-s + (1.65 − 2.87i)11-s + (−2.99 + 1.73i)12-s + 19.5i·13-s + (−9.58 + 2.45i)14-s + (−2.00 − 3.46i)16-s + (7.30 + 4.21i)17-s + (2.12 − 3.67i)18-s + (0.704 − 0.406i)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 + (0.269 − 0.963i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (0.150 − 0.260i)11-s + (−0.249 + 0.144i)12-s + 1.50i·13-s + (−0.684 + 0.175i)14-s + (−0.125 − 0.216i)16-s + (0.429 + 0.247i)17-s + (0.117 − 0.204i)18-s + (0.0370 − 0.0214i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.145i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.989 + 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.989 + 0.145i$
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: \((2,\ 1050,\ (\ :1),\ 0.989 + 0.145i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.956990062\)
\(L(\frac12)\) \(\approx\) \(1.956990062\)
\(L(2)\) 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 \( 1 + (-1.88 + 6.74i)T \)
good11 \( 1 + (-1.65 + 2.87i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 19.5iT - 169T^{2} \)
17 \( 1 + (-7.30 - 4.21i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-0.704 + 0.406i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-5.98 - 10.3i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 4.68T + 841T^{2} \)
31 \( 1 + (-27.9 - 16.1i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (3.01 + 5.21i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 19.6iT - 1.68e3T^{2} \)
43 \( 1 + 2.53T + 1.84e3T^{2} \)
47 \( 1 + (15.8 - 9.14i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-15.9 + 27.6i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-64.3 - 37.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-95.1 + 54.9i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (47.3 - 81.9i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 98.0T + 5.04e3T^{2} \)
73 \( 1 + (9.62 + 5.55i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (0.500 + 0.867i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 66.8iT - 6.88e3T^{2} \)
89 \( 1 + (-133. + 77.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 132. iT - 9.40e3T^{2} \)
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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

−9.797960517149978646842037941821, −8.943419938844163724765396187436, −8.258295714430605015431759980915, −7.33111932118090742509632521923, −6.57487899036751435246635779854, −5.06467630549352596287267299014, −4.13272648286184821953386876960, −3.47542074162591612023334386315, −2.14576997815723873133907546934, −1.04291404735233927872442745669, 0.815077286626769900811265313588, 2.26674156052718048849022483168, 3.27224230888561936775132405182, 4.73737034316792765503527143499, 5.57970647741992267670193405544, 6.39271114190961282633153911112, 7.42074669786918546511907152801, 8.167644337891674461917179090674, 8.648705918042430707844353706027, 9.634118055395700218699094486789

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