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Properties

Label 2-1950-13.10-c1-0-42
Degree $2$
Conductor $1950$
Sign $-0.921 - 0.387i$
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 + (0.749 + 0.432i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (0.151 − 0.0874i)11-s − 0.999·12-s + (−3.34 − 1.35i)13-s + 0.865·14-s + (−0.5 − 0.866i)16-s + (−4.08 + 7.08i)17-s + 0.999i·18-s + (−5.20 − 3.00i)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 + (0.283 + 0.163i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.0456 − 0.0263i)11-s − 0.288·12-s + (−0.926 − 0.375i)13-s + 0.231·14-s + (−0.125 − 0.216i)16-s + (−0.991 + 1.71i)17-s + 0.235i·18-s + (−1.19 − 0.689i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 - 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.921 - 0.387i$
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: \((2,\ 1950,\ (\ :1/2),\ -0.921 - 0.387i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6291328622\)
\(L(\frac12)\) \(\approx\) \(0.6291328622\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.34 + 1.35i)T \)
good7 \( 1 + (-0.749 - 0.432i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.151 + 0.0874i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (4.08 - 7.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.20 + 3.00i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.45 + 2.52i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.24 + 5.62i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.95iT - 31T^{2} \)
37 \( 1 + (-1.52 + 0.879i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.08 - 4.08i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.58 - 7.94i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 11.9iT - 47T^{2} \)
53 \( 1 + 2.48T + 53T^{2} \)
59 \( 1 + (-6.09 - 3.51i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.98 + 6.90i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.36 - 1.36i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-12.2 - 7.08i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 12.8iT - 73T^{2} \)
79 \( 1 + 9.48T + 79T^{2} \)
83 \( 1 + 0.139iT - 83T^{2} \)
89 \( 1 + (-11.3 + 6.56i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.48 + 4.32i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.505172910072638807808033838838, −8.090222673376280842532699243856, −6.88233097319216569471442340665, −6.35396197258126392651014430678, −5.52159802280319095714890040334, −4.59324452137199814911546590839, −3.89130556026241405098904044224, −2.45201082567212062341054699375, −1.89841667845188042641058893477, −0.17042252445382834671199520316, 1.91994303948582243142866101005, 3.05755690870974486394960422417, 4.09318927449226179811658622905, 4.86254307313987565199508721689, 5.35267446153942934785501576448, 6.56215254053438823985416100834, 7.03106633708668702895408196865, 7.945026201049029660954384266396, 8.924696384578938236272873566831, 9.511017493222334882473410748481

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