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Properties

Label 2-1950-65.29-c1-0-5
Degree $2$
Conductor $1950$
Sign $-0.450 - 0.892i$
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 + (0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (−0.499 + 0.866i)6-s + (−3.08 − 1.78i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (0.280 + 0.486i)11-s − 0.999i·12-s + (2.21 + 2.84i)13-s + 3.56·14-s + (−0.5 − 0.866i)16-s + (−2.70 − 1.56i)17-s + 0.999i·18-s + (−1.21 + 2.11i)19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.204 + 0.353i)6-s + (−1.16 − 0.673i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (0.0846 + 0.146i)11-s − 0.288i·12-s + (0.615 + 0.788i)13-s + 0.951·14-s + (−0.125 − 0.216i)16-s + (−0.655 − 0.378i)17-s + 0.235i·18-s + (−0.279 + 0.484i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.450 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.450 - 0.892i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (1849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ -0.450 - 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6227794046\)
\(L(\frac12)\) \(\approx\) \(0.6227794046\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
13 \( 1 + (-2.21 - 2.84i)T \)
good7 \( 1 + (3.08 + 1.78i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.280 - 0.486i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.70 + 1.56i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.21 - 2.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.65 - 3.84i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.561 - 0.972i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (0.486 - 0.280i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.56 - 2.70i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.379 + 0.219i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 4iT - 47T^{2} \)
53 \( 1 - 4.24iT - 53T^{2} \)
59 \( 1 + (5.12 - 8.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.842 + 1.45i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.2 - 5.90i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.28 - 10.8i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 9iT - 73T^{2} \)
79 \( 1 - 15.8T + 79T^{2} \)
83 \( 1 - 6.56iT - 83T^{2} \)
89 \( 1 + (-5.12 - 8.87i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.43 + 1.40i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.402224419967454017890468836152, −8.682114132324882887804242342005, −7.88506251112559819500733962927, −7.07130578514192105113977779911, −6.51787536419558989582973963020, −5.83099601965744844415077144382, −4.34834518439174235876187549216, −3.62631570011762295806000923973, −2.45506879343894802214049667486, −1.27837314520625049062211265122, 0.26237931149579169125726880491, 2.02125978727253124704450263963, 2.90991550119530379275506482430, 3.63587097568809932978416494472, 4.67175035597254460191089961477, 6.10639559065377200095118778295, 6.38810733557607793711949154827, 7.65966139098386517735810431289, 8.383153858746947334511934895727, 8.978251306714591995191712332727

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