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Properties

Label 2-1050-35.12-c1-0-3
Degree $2$
Conductor $1050$
Sign $0.310 - 0.950i$
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  + (−0.965 + 0.258i)2-s + (0.258 − 0.965i)3-s + (0.866 − 0.499i)4-s + i·6-s + (0.942 + 2.47i)7-s + (−0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (1.55 + 2.69i)11-s + (−0.258 − 0.965i)12-s + (−3.40 − 3.40i)13-s + (−1.55 − 2.14i)14-s + (0.500 − 0.866i)16-s + (5.14 + 1.37i)17-s + (0.965 + 0.258i)18-s + (−3.61 + 6.26i)19-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.149 − 0.557i)3-s + (0.433 − 0.249i)4-s + 0.408i·6-s + (0.356 + 0.934i)7-s + (−0.249 + 0.249i)8-s + (−0.288 − 0.166i)9-s + (0.468 + 0.811i)11-s + (−0.0747 − 0.278i)12-s + (−0.945 − 0.945i)13-s + (−0.414 − 0.572i)14-s + (0.125 − 0.216i)16-s + (1.24 + 0.334i)17-s + (0.227 + 0.0610i)18-s + (−0.829 + 1.43i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.310 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.310 - 0.950i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.310 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.041588715\)
\(L(\frac12)\) \(\approx\) \(1.041588715\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 \)
7 \( 1 + (-0.942 - 2.47i)T \)
good11 \( 1 + (-1.55 - 2.69i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.40 + 3.40i)T + 13iT^{2} \)
17 \( 1 + (-5.14 - 1.37i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.61 - 6.26i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.36 - 5.08i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 4.49iT - 29T^{2} \)
31 \( 1 + (7.98 - 4.61i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.47 - 0.929i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 2.51iT - 41T^{2} \)
43 \( 1 + (-3.86 + 3.86i)T - 43iT^{2} \)
47 \( 1 + (-1.05 - 3.94i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.32 - 0.890i)T + (45.8 + 26.5i)T^{2} \)
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 \( 1 + (-1.72 + 6.43i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 9.22T + 71T^{2} \)
73 \( 1 + (2.13 - 7.95i)T + (-63.2 - 36.5i)T^{2} \)
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

−9.901797407285885164693899733093, −9.257976493532893999237514512397, −8.233089747852562577417183944876, −7.76779332255619866655522416484, −6.95408708498951322497102479663, −5.80014957570963740064835930403, −5.29505876757781617424259293112, −3.66510028813360302384859692621, −2.37578716949144741804854534464, −1.45705042830228541666206855385, 0.59243007856238214903129272281, 2.18265280446959545586573620392, 3.42622610864778649359830641062, 4.35098328248377496657341322146, 5.31270311419412977708275320806, 6.70337662387891602915869660662, 7.26104811556612121726905141335, 8.274558410548956221635750854468, 9.067160886360142649777658934466, 9.642155980477920737484374647878

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