Properties

Label 2-1950-65.9-c1-0-10
Degree $2$
Conductor $1950$
Sign $0.187 - 0.982i$
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 + (0.486 − 0.280i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (−1.78 + 3.08i)11-s + 0.999i·12-s + (−1.35 + 3.34i)13-s − 0.561·14-s + (−0.5 + 0.866i)16-s + (4.43 − 2.56i)17-s − 0.999i·18-s + (−3.28 − 5.68i)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 + (0.183 − 0.106i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.536 + 0.929i)11-s + 0.288i·12-s + (−0.375 + 0.926i)13-s − 0.150·14-s + (−0.125 + 0.216i)16-s + (1.07 − 0.621i)17-s − 0.235i·18-s + (−0.752 − 1.30i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.187 - 0.982i$
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: \((2,\ 1950,\ (\ :1/2),\ 0.187 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.243528032\)
\(L(\frac12)\) \(\approx\) \(1.243528032\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.35 - 3.34i)T \)
good7 \( 1 + (-0.486 + 0.280i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.78 - 3.08i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-4.43 + 2.56i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.28 + 5.68i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.05 - 2.34i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.56 - 6.16i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-3.08 - 1.78i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.56 - 4.43i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.95 - 2.28i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 - 12.2iT - 53T^{2} \)
59 \( 1 + (-3.12 - 5.40i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.34 + 9.25i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.62 - 4.40i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.21 + 7.30i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 9iT - 73T^{2} \)
79 \( 1 + 4.80T + 79T^{2} \)
83 \( 1 + 2.43iT - 83T^{2} \)
89 \( 1 + (3.12 - 5.40i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-15.4 + 8.90i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.350168364392306596975404101332, −8.831987739926987579192566920598, −7.73471474779376766201372610149, −7.33752288221600516208926751722, −6.46097653062661807682270601955, −4.97263712311464545229678067944, −4.54389112367801155253328967450, −3.23056223218469422349534041556, −2.47954388237358839281174878466, −1.35885925759338636711128654148, 0.53144022772354775096773434576, 1.86927099160588701305880007077, 2.94947859657399609313782352910, 3.87651038779460206804680874286, 5.31321169561114887753763518686, 5.84509918482375604134510269476, 6.77540279793987286594799443354, 7.77312919886087245284090238272, 8.233684003172413059917806329954, 8.647445952103322557335362688620

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