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Properties

Label 2-1950-13.10-c1-0-29
Degree $2$
Conductor $1950$
Sign $0.711 + 0.702i$
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 + (−2.36 − 1.36i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (1.09 − 0.633i)11-s + 0.999·12-s + (2.59 + 2.5i)13-s − 2.73·14-s + (−0.5 − 0.866i)16-s + (2.86 − 4.96i)17-s + 0.999i·18-s + (4.09 + 2.36i)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 + (−0.894 − 0.516i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.331 − 0.191i)11-s + 0.288·12-s + (0.720 + 0.693i)13-s − 0.730·14-s + (−0.125 − 0.216i)16-s + (0.695 − 1.20i)17-s + 0.235i·18-s + (0.940 + 0.542i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.711 + 0.702i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.711 + 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.663684727\)
\(L(\frac12)\) \(\approx\) \(2.663684727\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.59 - 2.5i)T \)
good7 \( 1 + (2.36 + 1.36i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.09 + 0.633i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.86 + 4.96i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.09 - 2.36i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.09 + 3.63i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.23 - 3.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.46iT - 31T^{2} \)
37 \( 1 + (3.06 - 1.76i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-8.13 + 4.69i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.83 + 8.36i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.19iT - 47T^{2} \)
53 \( 1 - 6.46T + 53T^{2} \)
59 \( 1 + (6.92 + 4i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.59 + 7.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.3 + 6.56i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.09 - 2.36i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 6.26iT - 73T^{2} \)
79 \( 1 + 2.53T + 79T^{2} \)
83 \( 1 + 0.196iT - 83T^{2} \)
89 \( 1 + (8.19 - 4.73i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.19 - 3i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.353434522189262365709692347997, −8.458344366911216913579147138698, −7.33987731069870824434528837408, −6.67106704244770423842933119036, −5.76908142816943838228611766825, −4.93636391552560156415034608678, −3.82833653798591209144892445658, −3.48812757672656145818868770554, −2.38255743788492795765107849161, −0.876542969013062183631029622083, 1.24489510305999793323676883842, 2.67676563492584604856053156722, 3.37421997135436850283389383238, 4.24515032070989903854532850534, 5.67480403986000409170212217760, 5.93878941861164290816487413109, 6.82159928523467038648163272870, 7.70084204685919631894034527559, 8.288481343343367044184686460140, 9.227285373987011118011505939491

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