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Properties

Label 2-1050-35.9-c1-0-12
Degree $2$
Conductor $1050$
Sign $0.982 + 0.185i$
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  + (−0.866 − 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s − 0.999·6-s + (0.866 + 2.5i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (1 + 1.73i)11-s + (0.866 + 0.499i)12-s − 4i·13-s + (0.500 − 2.59i)14-s + (−0.5 + 0.866i)16-s + (4.33 − 2.5i)17-s + (−0.866 + 0.499i)18-s + (−2 + 3.46i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s − 0.408·6-s + (0.327 + 0.944i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (0.301 + 0.522i)11-s + (0.249 + 0.144i)12-s − 1.10i·13-s + (0.133 − 0.694i)14-s + (−0.125 + 0.216i)16-s + (1.05 − 0.606i)17-s + (−0.204 + 0.117i)18-s + (−0.458 + 0.794i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.982 + 0.185i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.982 + 0.185i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.541336464\)
\(L(\frac12)\) \(\approx\) \(1.541336464\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
7 \( 1 + (-0.866 - 2.5i)T \)
good11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 + (-4.33 + 2.5i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.33 + 2.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-5.5 - 9.52i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.92 - 4i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (0.866 + 0.5i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.3 + 6i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1 + 1.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 + T + 71T^{2} \)
73 \( 1 + (-1.73 + i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.5 + 7.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 + (-5.5 + 9.52i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - iT - 97T^{2} \)
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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

−10.00862762355920063609086138006, −8.964401516672035402492515581928, −8.233564932676098526040851030995, −7.79071287987323415232941472114, −6.64273474901885703925551548298, −5.71268431059938336224066880546, −4.56128744435200837603970440682, −3.20521289400795232920729482148, −2.43305850993258958500933393882, −1.17924432782220008145058168697, 1.01958224045922537387925486051, 2.38478080906580808555234399325, 3.85808522576099278482038013167, 4.50621518547548214069259727346, 5.89126898842557201679965533283, 6.68324806494428133621579849795, 7.71518440516929494161786861556, 8.155388634212292202407173274933, 9.173801483725241473417221321979, 9.774330960938632207652398901251

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