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Properties

Label 2-1050-35.17-c1-0-12
Degree $2$
Conductor $1050$
Sign $0.703 + 0.711i$
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  + (0.258 − 0.965i)2-s + (0.965 − 0.258i)3-s + (−0.866 − 0.499i)4-s − i·6-s + (2.47 + 0.942i)7-s + (−0.707 + 0.707i)8-s + (0.866 − 0.499i)9-s + (1.55 − 2.69i)11-s + (−0.965 − 0.258i)12-s + (3.40 + 3.40i)13-s + (1.55 − 2.14i)14-s + (0.500 + 0.866i)16-s + (1.37 + 5.14i)17-s + (−0.258 − 0.965i)18-s + (3.61 + 6.26i)19-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (0.557 − 0.149i)3-s + (−0.433 − 0.249i)4-s − 0.408i·6-s + (0.934 + 0.356i)7-s + (−0.249 + 0.249i)8-s + (0.288 − 0.166i)9-s + (0.468 − 0.811i)11-s + (−0.278 − 0.0747i)12-s + (0.945 + 0.945i)13-s + (0.414 − 0.572i)14-s + (0.125 + 0.216i)16-s + (0.334 + 1.24i)17-s + (−0.0610 − 0.227i)18-s + (0.829 + 1.43i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 + 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.703 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.703 + 0.711i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.703 + 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.420682790\)
\(L(\frac12)\) \(\approx\) \(2.420682790\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 \)
7 \( 1 + (-2.47 - 0.942i)T \)
good11 \( 1 + (-1.55 + 2.69i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.40 - 3.40i)T + 13iT^{2} \)
17 \( 1 + (-1.37 - 5.14i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.61 - 6.26i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.08 + 1.36i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 4.49iT - 29T^{2} \)
31 \( 1 + (7.98 + 4.61i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.929 + 3.47i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 2.51iT - 41T^{2} \)
43 \( 1 + (-3.86 + 3.86i)T - 43iT^{2} \)
47 \( 1 + (-3.94 - 1.05i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.890 + 3.32i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.666 - 1.15i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.8 + 6.27i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.43 - 1.72i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 9.22T + 71T^{2} \)
73 \( 1 + (7.95 - 2.13i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.65 - 0.957i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.97 + 8.97i)T + 83iT^{2} \)
89 \( 1 + (2.03 + 3.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.69 - 2.69i)T - 97iT^{2} \)
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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

−9.813281683523364023973222091909, −8.912595586327998585345487428588, −8.339011665429922852078120869554, −7.61419098234288460479978510685, −6.11222563749465366002132316979, −5.62610693248801258908291890550, −3.95341776757851507693748921803, −3.81094414349750542483049662083, −2.14931256193150435046382779494, −1.39649936871584979707984865136, 1.25586829823744035072522218577, 2.86969035466704855928611333136, 3.93687060958722056885290912570, 4.88193072236697573742932354393, 5.56429925977371251727885223297, 6.98029952053023538396194134673, 7.42882673239321175216329945911, 8.271713221451575382266283343003, 9.073502855327140826483686049376, 9.776115355820004549379921830346

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