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Properties

Label 2-1950-13.9-c1-0-4
Degree $2$
Conductor $1950$
Sign $-0.477 - 0.878i$
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 + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.499 − 0.866i)6-s + (−1 − 1.73i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−1 + 1.73i)11-s + 0.999·12-s + (−2.5 − 2.59i)13-s + 1.99·14-s + (−0.5 + 0.866i)16-s + (2.5 + 4.33i)17-s + 0.999·18-s + (1 + 1.73i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.204 − 0.353i)6-s + (−0.377 − 0.654i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.301 + 0.522i)11-s + 0.288·12-s + (−0.693 − 0.720i)13-s + 0.534·14-s + (−0.125 + 0.216i)16-s + (0.606 + 1.05i)17-s + 0.235·18-s + (0.229 + 0.397i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.477 - 0.878i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ -0.477 - 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8618366792\)
\(L(\frac12)\) \(\approx\) \(0.8618366792\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 \)
13 \( 1 + (2.5 + 2.59i)T \)
good7 \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.5 - 4.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.5 + 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (5.5 - 9.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.5 - 4.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5 - 8.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 2T + 47T^{2} \)
53 \( 1 - T + 53T^{2} \)
59 \( 1 + (-4 - 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.5 - 9.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7 - 12.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 13T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (1 - 1.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.677788571556883400528632144562, −8.477280762586459727543210948882, −7.956542057082898298864694613599, −7.05826323122888421135468924390, −6.35132326628754040069469738147, −5.47528181638312433633260009252, −4.68933852967818017784298570556, −3.83959236379475566361257914814, −2.66949632228976515784812760304, −0.991259587611552117330735909982, 0.45308523283961987254618226267, 1.85294822261728006529051524677, 2.80236306209461528046208106546, 3.63749792718024526195840527056, 5.15230332391603957313320796344, 5.45293530119668154936591671208, 6.88412999825637603769377373884, 7.23139481370337506647607902365, 8.259420724833814489426189103957, 9.237522268723670292361054805902

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