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Properties

Label 2-1950-13.10-c1-0-23
Degree $2$
Conductor $1950$
Sign $-0.252 + 0.967i$
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 + (−2.59 − 1.5i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (0.232 − 0.133i)11-s − 0.999·12-s + (−0.866 + 3.5i)13-s + 3·14-s + (−0.5 − 0.866i)16-s + (2 − 3.46i)17-s − 0.999i·18-s + (4.96 + 2.86i)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 + (−0.981 − 0.566i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.0699 − 0.0403i)11-s − 0.288·12-s + (−0.240 + 0.970i)13-s + 0.801·14-s + (−0.125 − 0.216i)16-s + (0.485 − 0.840i)17-s − 0.235i·18-s + (1.13 + 0.657i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.252 + 0.967i$
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: \((2,\ 1950,\ (\ :1/2),\ -0.252 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6585388038\)
\(L(\frac12)\) \(\approx\) \(0.6585388038\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 3.5i)T \)
good7 \( 1 + (2.59 + 1.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.232 + 0.133i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.96 - 2.86i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.73 - 3i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.732 + 1.26i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.92iT - 31T^{2} \)
37 \( 1 + (-5.13 + 2.96i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.46 - 2i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3 + 5.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.46iT - 47T^{2} \)
53 \( 1 - 0.267T + 53T^{2} \)
59 \( 1 + (9.92 + 5.73i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.267 + 0.464i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.26 - 0.732i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (11.1 + 6.46i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 - 3.07T + 79T^{2} \)
83 \( 1 - 9.46iT - 83T^{2} \)
89 \( 1 + (12.2 - 7.06i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.26 + 4.19i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.174989040882424135667088228363, −7.928931461257202058778295760467, −7.35568467034022838259069861851, −6.78021117206410320374116517634, −5.97708714276224548819047711842, −5.17611494431762045748514054102, −3.94223996729133162098315367672, −2.89163106274685130148448249652, −1.57295866413592374757341796826, −0.35659063836957216344422450850, 1.11332001141309890380702931459, 2.83673687110626820332320074019, 3.20621140500830928213433889845, 4.47351803992576777438824091026, 5.51143858477030856745526484842, 6.19989001941826674503214848193, 7.12579796261457636980686794837, 8.001747714756305907503107708418, 8.863797657227613346180421858342, 9.482206880526520964937276923429

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