Properties

Label 2-1950-13.10-c1-0-10
Degree $2$
Conductor $1950$
Sign $-0.00560 - 0.999i$
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 + (3.10 + 1.79i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (−4.72 + 2.72i)11-s + 0.999·12-s + (0.0664 + 3.60i)13-s + 3.59·14-s + (−0.5 − 0.866i)16-s + (−3.65 + 6.33i)17-s + 0.999i·18-s + (−3.34 − 1.93i)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 + (1.17 + 0.678i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (−1.42 + 0.821i)11-s + 0.288·12-s + (0.0184 + 0.999i)13-s + 0.959·14-s + (−0.125 − 0.216i)16-s + (−0.886 + 1.53i)17-s + 0.235i·18-s + (−0.767 − 0.443i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00560 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00560 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.00560 - 0.999i$
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: \((2,\ 1950,\ (\ :1/2),\ -0.00560 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.315113317\)
\(L(\frac12)\) \(\approx\) \(2.315113317\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0664 - 3.60i)T \)
good7 \( 1 + (-3.10 - 1.79i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.72 - 2.72i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.65 - 6.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.34 + 1.93i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.929 + 1.60i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.67 - 4.63i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.23iT - 31T^{2} \)
37 \( 1 + (1.84 - 1.06i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.33 - 3.65i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.10 + 7.10i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.07iT - 47T^{2} \)
53 \( 1 - 2.23T + 53T^{2} \)
59 \( 1 + (0.237 + 0.137i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.06 + 3.57i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-12.8 + 7.39i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-12.3 - 7.14i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 14.8iT - 73T^{2} \)
79 \( 1 - 2.62T + 79T^{2} \)
83 \( 1 + 7.15iT - 83T^{2} \)
89 \( 1 + (-8.30 + 4.79i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.65 + 0.954i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.374843753790461738181075602294, −8.548147022123338056038807495821, −8.062715036040318825886875116256, −6.92868988247078626707693929666, −6.03077969112989172168595135441, −4.85596875416811799692225302339, −4.75836459756206583226299503103, −3.68512303158381468866155662027, −2.22576999742039588665110815640, −2.02155111823987375400062593304, 0.59995812923965532096450743672, 2.19627501313883426662666811924, 2.99532377156194164089562990831, 4.13145422958736870249162925191, 5.08640316635253539115051459695, 5.58872581513551157293781345504, 6.74525257745797967881483876038, 7.45130685310649848097794773422, 8.228499018934894823868170012320, 8.393921811215392983590262323066

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