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Properties

Label 2-1950-65.4-c1-0-27
Degree $2$
Conductor $1950$
Sign $-0.215 + 0.976i$
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 + (0.140 + 0.242i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (0.663 + 0.383i)11-s + 0.999i·12-s + (−2.30 + 2.77i)13-s + 0.280·14-s + (−0.5 + 0.866i)16-s + (1.92 − 1.10i)17-s + 0.999·18-s + (4.57 − 2.63i)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 + (0.0529 + 0.0917i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (0.200 + 0.115i)11-s + 0.288i·12-s + (−0.639 + 0.769i)13-s + 0.0749·14-s + (−0.125 + 0.216i)16-s + (0.465 − 0.269i)17-s + 0.235·18-s + (1.04 − 0.605i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.215 + 0.976i$
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: \((2,\ 1950,\ (\ :1/2),\ -0.215 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.612384833\)
\(L(\frac12)\) \(\approx\) \(1.612384833\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.30 - 2.77i)T \)
good7 \( 1 + (-0.140 - 0.242i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.663 - 0.383i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.92 + 1.10i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.57 + 2.63i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.25 - 0.725i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.03 + 5.24i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.28iT - 31T^{2} \)
37 \( 1 + (-1.77 + 3.07i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.92 - 1.10i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.12 - 1.80i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.06T + 47T^{2} \)
53 \( 1 + 3.28iT - 53T^{2} \)
59 \( 1 + (-4.32 + 2.49i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.773 + 1.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.71 + 6.43i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.09 - 1.21i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 14.2T + 73T^{2} \)
79 \( 1 - 14.4T + 79T^{2} \)
83 \( 1 - 4.42T + 83T^{2} \)
89 \( 1 + (-4.60 - 2.65i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.633 - 1.09i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.324656218393344357060176043498, −8.135288546887219830396385944550, −7.27491703255302082407664471639, −6.55362890633605430669221749873, −5.59139815407145108008887871089, −4.89606044822943812853939763321, −4.06519991124839950253684714530, −2.90515647626044407240865403271, −1.93776797085197480707551017317, −0.67371513206286518083900684849, 1.09224009111002033080390867803, 2.87848397171140346901009777207, 3.70951835965239638512088898561, 4.77350266480699920043911592630, 5.39345677070208241222452593688, 6.10161612081296862819587209233, 7.04861754756334616781166894930, 7.67556659993706173363653175956, 8.524205264549347658026410280595, 9.390253751641201904677185642916

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