Properties

Label 2-1950-65.9-c1-0-20
Degree $2$
Conductor $1950$
Sign $0.758 + 0.651i$
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 + (2.73 − 1.58i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (−1.5 + 2.59i)11-s − 0.999i·12-s + (3.60 − 0.0811i)13-s − 3.16·14-s + (−0.5 + 0.866i)16-s + (0.725 − 0.418i)17-s − 0.999i·18-s + (−1.58 − 2.73i)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 + (1.03 − 0.597i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.452 + 0.783i)11-s − 0.288i·12-s + (0.999 − 0.0225i)13-s − 0.845·14-s + (−0.125 + 0.216i)16-s + (0.175 − 0.101i)17-s − 0.235i·18-s + (−0.362 − 0.628i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.758 + 0.651i$
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: \((2,\ 1950,\ (\ :1/2),\ 0.758 + 0.651i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.258614844\)
\(L(\frac12)\) \(\approx\) \(1.258614844\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.60 + 0.0811i)T \)
good7 \( 1 + (-2.73 + 1.58i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.725 + 0.418i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.58 + 2.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.87 - 1.08i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.16 + 3.74i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.83T + 31T^{2} \)
37 \( 1 + (-7.34 - 4.24i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.74 - 8.21i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.00 + 0.581i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 + 0.837iT - 53T^{2} \)
59 \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.24 - 3.88i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.46 + 2i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.08 - 12.2i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 11.3iT - 73T^{2} \)
79 \( 1 + 2.83T + 79T^{2} \)
83 \( 1 + 11.6iT - 83T^{2} \)
89 \( 1 + (0.418 - 0.725i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-15.0 + 8.66i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.110837430765287683019890195552, −8.112906983158558170463670606093, −7.77825980972248395943757799664, −6.85109101234599545759501775895, −6.08600407412214746522012820939, −4.85469969107873577373758485653, −4.33624642610928213918207960827, −2.96089064787993921546580617378, −1.78262166114189413941131189005, −0.875085152751153338471966518835, 0.900136242030444701118745257303, 2.09222947394439593141247025137, 3.43790482327564088230284975164, 4.59328621625424199855580526004, 5.50356717839527744192456748065, 5.96179747995102462884665450129, 6.90238289534423665412352591196, 7.965697158074900524866516886148, 8.487293574314402663196751389144, 9.036623379831513009753062930263

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