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Properties

Label 2-1950-65.4-c1-0-25
Degree $2$
Conductor $1950$
Sign $0.00837 + 0.999i$
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  + (0.5 − 0.866i)2-s + (−0.866 − 0.5i)3-s + (−0.499 − 0.866i)4-s + (−0.866 + 0.499i)6-s + (0.633 + 1.09i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (−1.09 − 0.633i)11-s + 0.999i·12-s + (3.23 − 1.59i)13-s + 1.26·14-s + (−0.5 + 0.866i)16-s + (−4.5 + 2.59i)17-s + 0.999·18-s + (4.09 − 2.36i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.353 + 0.204i)6-s + (0.239 + 0.415i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (−0.331 − 0.191i)11-s + 0.288i·12-s + (0.896 − 0.443i)13-s + 0.338·14-s + (−0.125 + 0.216i)16-s + (−1.09 + 0.630i)17-s + 0.235·18-s + (0.940 − 0.542i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00837 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00837 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.00837 + 0.999i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.00837 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.728476595\)
\(L(\frac12)\) \(\approx\) \(1.728476595\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 \)
13 \( 1 + (-3.23 + 1.59i)T \)
good7 \( 1 + (-0.633 - 1.09i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.09 + 0.633i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (4.5 - 2.59i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.09 + 2.36i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-7.09 - 4.09i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.46iT - 31T^{2} \)
37 \( 1 + (-1.5 + 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.59 - 3.23i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.63 - 2.09i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.73T + 47T^{2} \)
53 \( 1 + 3iT - 53T^{2} \)
59 \( 1 + (-12 + 6.92i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.59 + 13.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.63 - 6.29i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.90 + 1.09i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 12.1T + 73T^{2} \)
79 \( 1 + 8.39T + 79T^{2} \)
83 \( 1 + 5.66T + 83T^{2} \)
89 \( 1 + (-8.19 - 4.73i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (3 + 5.19i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.087568549960199432936103855144, −8.294436978191395994260510287000, −7.38031285456726670229852904857, −6.43464262524988442818737724412, −5.62550279266785482711479197376, −5.05333550475732878482386521548, −3.98723087260874925067378693124, −3.00102504570790996074387821184, −1.94507736298041121853131279670, −0.75083650722170635970574602903, 1.05688471777580820580716598283, 2.72855789663734331184171696247, 3.85242805544941582113810774158, 4.63498196904425243087019416193, 5.29685853754478564160501263475, 6.22605211320916287812098274568, 6.97808705743515482778261963456, 7.55160894855942862707553501039, 8.737473507879877908600591703501, 9.093975712191692578297101110246

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