Properties

Label 2-1950-13.10-c1-0-24
Degree $2$
Conductor $1950$
Sign $0.505 + 0.862i$
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 + (0.807 + 0.465i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (1.45 − 0.841i)11-s − 0.999·12-s + (1.86 + 3.08i)13-s + 0.931·14-s + (−0.5 − 0.866i)16-s + (1.27 − 2.21i)17-s + 0.999i·18-s + (2.27 + 1.31i)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 + (0.305 + 0.176i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.439 − 0.253i)11-s − 0.288·12-s + (0.516 + 0.856i)13-s + 0.249·14-s + (−0.125 − 0.216i)16-s + (0.309 − 0.536i)17-s + 0.235i·18-s + (0.522 + 0.301i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 + 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.505 + 0.862i$
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: \((2,\ 1950,\ (\ :1/2),\ 0.505 + 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.567462271\)
\(L(\frac12)\) \(\approx\) \(2.567462271\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.86 - 3.08i)T \)
good7 \( 1 + (-0.807 - 0.465i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.45 + 0.841i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.27 + 2.21i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.27 - 1.31i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.22 - 5.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.69 - 2.93i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.29iT - 31T^{2} \)
37 \( 1 + (-7.93 + 4.57i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.01 + 0.587i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.47 - 2.55i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.97iT - 47T^{2} \)
53 \( 1 + 5.36T + 53T^{2} \)
59 \( 1 + (3.44 + 1.98i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.36 + 7.55i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.49 + 2.59i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.40 - 1.96i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 7.26iT - 73T^{2} \)
79 \( 1 + 3.68T + 79T^{2} \)
83 \( 1 - 3.84iT - 83T^{2} \)
89 \( 1 + (-11.6 + 6.73i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-12.1 - 7.04i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.284058058712416629387238632858, −8.141433078057995524295263896753, −7.39212358849537867323630184177, −6.53335222478951148249081127494, −5.82415339687378933137691840910, −5.05676587249203855476292902787, −4.08860963960692145157645453428, −3.16571397386776838900763973304, −1.99718990694510341553815054844, −1.02097769590787542042763719834, 1.12426598978256441994374222072, 2.76761927562030608941989602013, 3.62107009830625798373906178766, 4.57013037537914071800069055953, 5.15880772720175633822756967597, 6.12248485724357732137312346827, 6.72355768152238189771014293973, 7.74565450161725971119417506672, 8.438517323358282152218110199898, 9.242753032112663141945651500324

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