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Properties

Label 2-1950-13.4-c1-0-33
Degree $2$
Conductor $1950$
Sign $0.862 + 0.505i$
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 + (−0.417 + 0.241i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + (−2.00 − 1.15i)11-s + 0.999·12-s + (3.08 + 1.86i)13-s − 0.482·14-s + (−0.5 + 0.866i)16-s + (−3.39 − 5.88i)17-s − 0.999i·18-s + (4.39 − 2.53i)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 + (−0.157 + 0.0911i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.605 − 0.349i)11-s + 0.288·12-s + (0.856 + 0.516i)13-s − 0.128·14-s + (−0.125 + 0.216i)16-s + (−0.824 − 1.42i)17-s − 0.235i·18-s + (1.00 − 0.582i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 + 0.505i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.862 + 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.862 + 0.505i$
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: \((2,\ 1950,\ (\ :1/2),\ 0.862 + 0.505i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.689568858\)
\(L(\frac12)\) \(\approx\) \(2.689568858\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.08 - 1.86i)T \)
good7 \( 1 + (0.417 - 0.241i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.00 + 1.15i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.39 + 5.88i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.39 + 2.53i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.45 + 5.98i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.20 + 7.28i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.952iT - 31T^{2} \)
37 \( 1 + (-5.35 - 3.09i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.55 - 4.36i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.64 - 9.78i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 5.77iT - 47T^{2} \)
53 \( 1 + 5.81T + 53T^{2} \)
59 \( 1 + (-10.2 + 5.91i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.68 + 8.11i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.40 + 3.69i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.01 + 1.74i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 8.53iT - 73T^{2} \)
79 \( 1 - 9.03T + 79T^{2} \)
83 \( 1 + 7.94iT - 83T^{2} \)
89 \( 1 + (-7.80 - 4.50i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.95 - 4.59i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.128841713324490354420731281661, −8.059641575683315391445725333255, −7.61766231093160216708578078240, −6.45819248371701286939589851934, −6.28085283346800347918835454291, −4.96131508080787258946815039908, −4.38061875260404049798942325331, −2.98043647897242725338414670787, −2.57089586250238804538209269826, −0.842015704321164686137931263198, 1.32434351704705594652790319960, 2.56056671032131653124825104152, 3.54799182654846135466837915891, 4.08505833563717494282433198898, 5.29001932357851180353863666364, 5.71702748290317275944764006421, 6.84244706786261996528664898403, 7.69013442853283653982755444447, 8.598601217505405610225828536885, 9.282676054047654648488648766558

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