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Properties

Label 2-1950-13.4-c1-0-23
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: $\chi_{1950} (901, \cdot )$
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

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

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