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Properties

Label 2-1050-35.9-c1-0-1
Degree $2$
Conductor $1050$
Sign $-0.192 - 0.981i$
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 + (−1.73 − 2i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.866 − 0.499i)12-s + i·13-s + (0.499 + 2.59i)14-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.499i)18-s + (−1.5 + 2.59i)19-s + (2.5 + 0.866i)21-s − 0.999i·22-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 + (−0.654 − 0.755i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (0.150 + 0.261i)11-s + (−0.249 − 0.144i)12-s + 0.277i·13-s + (0.133 + 0.694i)14-s + (−0.125 + 0.216i)16-s + (−0.204 + 0.117i)18-s + (−0.344 + 0.596i)19-s + (0.545 + 0.188i)21-s − 0.213i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.192 - 0.981i$
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: \((2,\ 1050,\ (\ :1/2),\ -0.192 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4347871061\)
\(L(\frac12)\) \(\approx\) \(0.4347871061\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 + 2i)T \)
good11 \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - iT - 13T^{2} \)
17 \( 1 + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.5 - 2.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.06 + 3.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-9.52 - 5.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 11T + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 + (4.33 + 2.5i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (9.52 - 5.5i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (5.19 - 3i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 8iT - 83T^{2} \)
89 \( 1 + (5 - 8.66i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 16iT - 97T^{2} \)
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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

−10.04007331385252304572081229304, −9.711896020228065602674908338159, −8.520167814969533306583362223686, −7.79540939934468780821060490540, −6.61626681642642048571109342739, −6.27247098058301995790817486838, −4.71165555941424503048609533944, −3.96160036411659492744353436473, −2.81298008647284192148154497594, −1.26861404259426359437036417380, 0.27335889485403576172885470641, 1.93264561848697945462907098021, 3.15955263528094768091559034885, 4.63722305760108412589690753785, 5.75560105492558559235404173295, 6.25656280712652071989658912377, 7.10673217437251471965409218023, 8.104171908779820647803368011115, 8.772509503233306378477662116910, 9.730032836443808680045021894207

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