Properties

Label 2-1950-13.4-c1-0-29
Degree $2$
Conductor $1950$
Sign $0.998 - 0.0575i$
Analytic cond. $15.5708$
Root an. cond. $3.94598$
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 + (0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + (0.866 − 0.499i)6-s + (2.01 − 1.16i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + (4.62 + 2.67i)11-s + 0.999·12-s + (3.60 − 0.161i)13-s + 2.32·14-s + (−0.5 + 0.866i)16-s + (−2 − 3.46i)17-s − 0.999i·18-s + (−3.48 + 2.01i)19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s + (0.353 − 0.204i)6-s + (0.760 − 0.438i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (1.39 + 0.805i)11-s + 0.288·12-s + (0.998 − 0.0447i)13-s + 0.620·14-s + (−0.125 + 0.216i)16-s + (−0.485 − 0.840i)17-s − 0.235i·18-s + (−0.799 + 0.461i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0575i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.998 - 0.0575i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.998 - 0.0575i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.367827399\)
\(L(\frac12)\) \(\approx\) \(3.367827399\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 \)
13 \( 1 + (-3.60 + 0.161i)T \)
good7 \( 1 + (-2.01 + 1.16i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.62 - 2.67i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.48 - 2.01i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.46 + 4.27i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.14 - 3.71i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.47iT - 31T^{2} \)
37 \( 1 + (-2.72 - 1.57i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.29 - 1.32i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.12 + 10.6i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 1.81iT - 47T^{2} \)
53 \( 1 - 5.48T + 53T^{2} \)
59 \( 1 + (-5.87 + 3.39i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.267 + 0.464i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.55 + 2.05i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-13.7 + 7.93i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 13.5iT - 73T^{2} \)
79 \( 1 + 7.96T + 79T^{2} \)
83 \( 1 - 11.3iT - 83T^{2} \)
89 \( 1 + (1.50 + 0.869i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-13.9 + 8.05i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.794132807948302995286965089119, −8.497708635268727596954776119716, −7.37582167364683989497038062959, −6.84041069194023775608012550881, −6.22544587005023736350445555568, −5.04744672088102266430501738860, −4.26874198633837341920880250113, −3.54196643674786248372702769150, −2.20336296424896633306526608972, −1.23335174190037613221964578843, 1.28186138956945485334576995110, 2.30236746574281210099823021568, 3.58612472910013080188858331927, 4.04348838569429981502337350073, 4.98118616286302946897616224083, 6.01155274000084695633810890954, 6.42637117482388162147394969261, 7.74835479125710771653817809039, 8.707702412310714438891378305204, 8.982209733990414940484072847113

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