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Properties

Label 2-1950-65.4-c1-0-19
Degree $2$
Conductor $1950$
Sign $0.926 + 0.376i$
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  + (−0.5 + 0.866i)2-s + (−0.866 − 0.5i)3-s + (−0.499 − 0.866i)4-s + (0.866 − 0.499i)6-s + (−2.36 − 4.09i)7-s + 0.999·8-s + (0.499 + 0.866i)9-s + (4.09 + 2.36i)11-s + 0.999i·12-s + (0.232 + 3.59i)13-s + 4.73·14-s + (−0.5 + 0.866i)16-s + (4.5 − 2.59i)17-s − 0.999·18-s + (−1.09 + 0.633i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.353 − 0.204i)6-s + (−0.894 − 1.54i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (1.23 + 0.713i)11-s + 0.288i·12-s + (0.0643 + 0.997i)13-s + 1.26·14-s + (−0.125 + 0.216i)16-s + (1.09 − 0.630i)17-s − 0.235·18-s + (−0.251 + 0.145i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.376i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 + 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.926 + 0.376i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.926 + 0.376i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.051798640\)
\(L(\frac12)\) \(\approx\) \(1.051798640\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 \)
13 \( 1 + (-0.232 - 3.59i)T \)
good7 \( 1 + (2.36 + 4.09i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.09 - 2.36i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-4.5 + 2.59i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.09 - 0.633i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.90 + 1.09i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.53iT - 31T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.401 - 0.232i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.36 + 3.09i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.26T + 47T^{2} \)
53 \( 1 + 3iT - 53T^{2} \)
59 \( 1 + (-12 + 6.92i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.40 + 4.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.36 + 9.29i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.09 + 4.09i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 12.1T + 73T^{2} \)
79 \( 1 - 12.3T + 79T^{2} \)
83 \( 1 + 11.6T + 83T^{2} \)
89 \( 1 + (2.19 + 1.26i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3 - 5.19i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.420258365713734504812222219261, −8.167649184173291132643043013854, −7.28790159566739027906259251633, −6.77144341889434562095372723233, −6.41384542843255476010328739160, −5.18337641443156932935946125744, −4.22674022738484233140536031448, −3.56821306658274984288432855440, −1.70598090187077076039991618599, −0.65257488186604479019493813770, 0.899895902825374714813872590449, 2.36053324898845470639438050208, 3.33767152909756901003176324416, 3.99937605032833357132516089858, 5.54174790402080838891615237606, 5.81532075590722478644480351391, 6.69025809877059780457286066386, 7.969348858532398374554499357468, 8.659947645627630116909979994589, 9.384028507150076104504480761300

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