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Properties

Label 2-1950-13.4-c1-0-34
Degree $2$
Conductor $1950$
Sign $-0.934 - 0.354i$
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 + (−4.02 + 2.32i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + (3.81 + 2.20i)11-s + 0.999·12-s + (3.35 + 1.32i)13-s + 4.64·14-s + (−0.5 + 0.866i)16-s + (−2 − 3.46i)17-s + 0.999i·18-s + (−6.96 + 4.02i)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 + (−1.52 + 0.877i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (1.14 + 0.663i)11-s + 0.288·12-s + (0.930 + 0.366i)13-s + 1.24·14-s + (−0.125 + 0.216i)16-s + (−0.485 − 0.840i)17-s + 0.235i·18-s + (−1.59 + 0.922i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.354i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.934 - 0.354i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ -0.934 - 0.354i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.008514078457\)
\(L(\frac12)\) \(\approx\) \(0.008514078457\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.35 - 1.32i)T \)
good7 \( 1 + (4.02 - 2.32i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.81 - 2.20i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.96 - 4.02i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.488 + 0.845i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.15 + 3.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.44iT - 31T^{2} \)
37 \( 1 + (3.28 + 1.89i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.31 + 3.64i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.358 + 0.620i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 9.75iT - 47T^{2} \)
53 \( 1 + 13.5T + 53T^{2} \)
59 \( 1 + (1.88 - 1.09i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.73 + 6.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.58 - 0.912i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.88 - 3.97i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 4.36iT - 73T^{2} \)
79 \( 1 + 14.9T + 79T^{2} \)
83 \( 1 - 3.51iT - 83T^{2} \)
89 \( 1 + (-7.07 - 4.08i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (11.9 - 6.91i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.914259013107300070461972895651, −8.218428793634418023505206414930, −7.10395481527072933587831631048, −6.35306913974865091122187733592, −6.10991906955989036300673136044, −4.33899674149941679496678823991, −3.51943283146633802286449440097, −2.52377102198216308430961333583, −1.66124831652593353105429360434, −0.00362038470875086783940575798, 1.42808195887286054505469173630, 3.10946322447399766636850382562, 3.69117230279806945662357127693, 4.63165617249803866724644482925, 6.06636372191636775788622226459, 6.51939066447331778726046424121, 7.07559967483456769689750257853, 8.570625714107036807387870917575, 8.634035495058018609602387509236, 9.497307722199500913270697907107

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