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Properties

Label 2-1950-13.10-c1-0-32
Degree $2$
Conductor $1950$
Sign $0.265 + 0.964i$
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 + (1.73 + i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (5.59 − 3.23i)11-s − 0.999·12-s + (−1 − 3.46i)13-s + 1.99·14-s + (−0.5 − 0.866i)16-s + (−2 + 3.46i)17-s + 0.999i·18-s + (6.46 + 3.73i)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 + (0.654 + 0.377i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (1.68 − 0.974i)11-s − 0.288·12-s + (−0.277 − 0.960i)13-s + 0.534·14-s + (−0.125 − 0.216i)16-s + (−0.485 + 0.840i)17-s + 0.235i·18-s + (1.48 + 0.856i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.265 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.265 + 0.964i$
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: \((2,\ 1950,\ (\ :1/2),\ 0.265 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.745684367\)
\(L(\frac12)\) \(\approx\) \(2.745684367\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 3.46i)T \)
good7 \( 1 + (-1.73 - i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.59 + 3.23i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.46 - 3.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.86 - 3.23i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.133 + 0.232i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
37 \( 1 + (7.96 - 4.59i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.73 + i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.96 + 10.3i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.53iT - 47T^{2} \)
53 \( 1 + 0.928T + 53T^{2} \)
59 \( 1 + (-7.33 - 4.23i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.19 + 9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.92 + 5.73i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (10.7 + 6.19i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + 13.9T + 79T^{2} \)
83 \( 1 - 8.92iT - 83T^{2} \)
89 \( 1 + (0.464 - 0.267i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.464 + 0.267i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.916009434412067100657964256199, −8.297069422449969991314577319981, −7.32135064349235588758604166765, −6.51200978063584328309741927471, −5.63300835585147496181358723958, −5.23413440786850984976479078668, −3.87980625413237533721753077235, −3.25805420995621112095734326806, −1.86199784760278347246423851262, −1.02495137654396046890362660467, 1.28987945045708213538269323309, 2.63099828136384666051847778020, 3.92423676106641521835894446190, 4.51936173342881687396599389587, 5.04467570339801850969280459349, 6.19948379860157797397760798500, 7.11763646335761427987545772825, 7.28214356915025855143346310111, 8.752616508717425790267398253795, 9.305181167774638055975378761140

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