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Properties

Label 2-1950-13.10-c1-0-38
Degree $2$
Conductor $1950$
Sign $-0.777 + 0.629i$
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 + (−0.242 − 0.140i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (0.663 − 0.383i)11-s − 0.999·12-s + (2.77 − 2.30i)13-s − 0.280·14-s + (−0.5 − 0.866i)16-s + (1.10 − 1.92i)17-s + 0.999i·18-s + (−4.57 − 2.63i)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 + (−0.0917 − 0.0529i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.200 − 0.115i)11-s − 0.288·12-s + (0.769 − 0.639i)13-s − 0.0749·14-s + (−0.125 − 0.216i)16-s + (0.269 − 0.465i)17-s + 0.235i·18-s + (−1.04 − 0.605i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.777 + 0.629i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.777 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.777 + 0.629i$
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: \((2,\ 1950,\ (\ :1/2),\ -0.777 + 0.629i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.880680157\)
\(L(\frac12)\) \(\approx\) \(1.880680157\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.77 + 2.30i)T \)
good7 \( 1 + (0.242 + 0.140i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.663 + 0.383i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.10 + 1.92i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.57 + 2.63i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.725 - 1.25i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.03 + 5.24i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.28iT - 31T^{2} \)
37 \( 1 + (-3.07 + 1.77i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.92 + 1.10i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.80 + 3.12i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.06iT - 47T^{2} \)
53 \( 1 + 3.28T + 53T^{2} \)
59 \( 1 + (4.32 + 2.49i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.773 - 1.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.43 + 3.71i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.09 + 1.21i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 14.2iT - 73T^{2} \)
79 \( 1 + 14.4T + 79T^{2} \)
83 \( 1 - 4.42iT - 83T^{2} \)
89 \( 1 + (4.60 - 2.65i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.09 + 0.633i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.879866032489663465009857162384, −8.026023420881303775576253091443, −7.16103734139141902985523405979, −6.34141049823171225654430866744, −5.72493451860391881107488852938, −4.82742788554034270709835657159, −3.85677006841069580047819911381, −2.91901482205567691476011270503, −1.84557625180891160849281380875, −0.57043145794947072796007676526, 1.58535405389491813289255506671, 2.95421166845949283263511363349, 3.99693645164207785011496626174, 4.44745663152593484003932870980, 5.60172627505576791013477563023, 6.18522915171380016238932677984, 6.88554927841268701519071308685, 7.936119333759359075807242120804, 8.673912494848916585022811988752, 9.430862857227590445022248679455

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