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Properties

Label 2-1950-13.10-c1-0-5
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 + (0.401 − 0.232i)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 + (−0.464 − 0.267i)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 + (0.121 − 0.0699i)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 + (−0.106 − 0.0614i)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: $\chi_{1950} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ -0.265 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3718411986\)
\(L(\frac12)\) \(\approx\) \(0.3718411986\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.401 + 0.232i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.464 + 0.267i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.133 - 0.232i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.86 + 3.23i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
37 \( 1 + (1.03 - 0.598i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.73 - i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.964 - 1.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 10.4iT - 47T^{2} \)
53 \( 1 - 12.9T + 53T^{2} \)
59 \( 1 + (1.33 + 0.767i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.19 - 9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.92 - 2.26i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.26 + 4.19i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 + 0.0717T + 79T^{2} \)
83 \( 1 - 4.92iT - 83T^{2} \)
89 \( 1 + (-6.46 + 3.73i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.46 - 3.73i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.337294403162206471632822852734, −8.549473870246104668795070356915, −7.78123059565408424367090898443, −7.13870872803815850785751799767, −6.30288734498247466651051855860, −5.77397555184264790586563006061, −4.67666284124574887771239654372, −3.48973585444318246460146564919, −2.36149326721524111922882274802, −1.05982964508909167620417089179, 0.19451709011601680327916292132, 1.89127148038880846063393944289, 2.92581257043853707049120219946, 3.89629049687294735001219729209, 4.80841350826342263292781785553, 5.80366352473998086450495290098, 6.75693075711420627702303961302, 7.27385877679319593281333610170, 8.531314944205767214496533073530, 9.107778937735019130027753548622

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