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Properties

Label 2-1950-65.9-c1-0-21
Degree $2$
Conductor $1950$
Sign $0.982 + 0.187i$
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 + (2.06 − 3.57i)11-s − 0.999i·12-s + (1.35 − 3.34i)13-s − 3.56·14-s + (−0.5 + 0.866i)16-s + (−4.43 + 2.56i)17-s + 0.999i·18-s + (−1.78 − 3.08i)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 + (0.621 − 1.07i)11-s − 0.288i·12-s + (0.375 − 0.926i)13-s − 0.951·14-s + (−0.125 + 0.216i)16-s + (−1.07 + 0.621i)17-s + 0.235i·18-s + (−0.408 − 0.707i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.187i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.982 + 0.187i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (1699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.982 + 0.187i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.780160217\)
\(L(\frac12)\) \(\approx\) \(1.780160217\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.35 + 3.34i)T \)
good7 \( 1 + (3.08 - 1.78i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.06 + 3.57i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (4.43 - 2.56i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.78 + 3.08i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.65 - 3.84i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.28 + 5.68i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.68T + 31T^{2} \)
37 \( 1 + (-3.57 - 2.06i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.12 + 3.67i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.95 + 2.28i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 7iT - 47T^{2} \)
53 \( 1 - 4.43iT - 53T^{2} \)
59 \( 1 + (-5.28 - 9.14i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-12.3 - 7.12i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.43 - 4.22i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 15.3iT - 73T^{2} \)
79 \( 1 + 7.43T + 79T^{2} \)
83 \( 1 - 1.12iT - 83T^{2} \)
89 \( 1 + (-0.903 + 1.56i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.972 - 0.561i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.887812560935922427660816893786, −8.508297606792610297960410059675, −7.33503906850662907052735131180, −6.42863107033293858876899320496, −6.14140497595238526512965329964, −5.37607643544604316464862309746, −4.27472756660577133447272041928, −3.28412584279150506615181350224, −2.51184252108664876834001200849, −0.70322667241907149127332760433, 1.01106139903692672474078142961, 2.41755616200565892754369870735, 3.52711322970781169105374422348, 4.39170521973504649722558090999, 4.80309213147211005861243114224, 6.24494153011246076732337958845, 6.65221065272230693576379007909, 7.17184241461247576083681623445, 8.681799137533102319359082144877, 9.522789402914728436577702557522

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