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Properties

Label 2-1950-13.10-c1-0-26
Degree $2$
Conductor $1950$
Sign $0.457 + 0.889i$
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 + (3.76 + 2.17i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (−2.04 + 1.17i)11-s − 0.999·12-s + (−1.69 − 3.18i)13-s − 4.34·14-s + (−0.5 − 0.866i)16-s + (1.50 − 2.60i)17-s − 0.999i·18-s + (−0.585 − 0.338i)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 + (1.42 + 0.821i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.615 + 0.355i)11-s − 0.288·12-s + (−0.469 − 0.883i)13-s − 1.16·14-s + (−0.125 − 0.216i)16-s + (0.364 − 0.631i)17-s − 0.235i·18-s + (−0.134 − 0.0776i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.457 + 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.457 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.457 + 0.889i$
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: \((2,\ 1950,\ (\ :1/2),\ 0.457 + 0.889i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.059267147\)
\(L(\frac12)\) \(\approx\) \(1.059267147\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 + 3.18i)T \)
good7 \( 1 + (-3.76 - 2.17i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.04 - 1.17i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.50 + 2.60i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.585 + 0.338i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.22 + 5.58i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.82 + 8.35i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.11iT - 31T^{2} \)
37 \( 1 + (-6.48 + 3.74i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.60 - 1.50i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.41 + 5.91i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.61iT - 47T^{2} \)
53 \( 1 - 9.43T + 53T^{2} \)
59 \( 1 + (-4.56 - 2.63i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.15 + 3.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.04 - 2.91i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.52 - 1.45i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 7.67iT - 73T^{2} \)
79 \( 1 - 3.74T + 79T^{2} \)
83 \( 1 - 10.3iT - 83T^{2} \)
89 \( 1 + (4.15 - 2.39i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-14.1 - 8.17i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.789951006391875206636038356106, −8.129312408820133335474030993816, −7.68912206558951555791202472304, −6.88891794065197221024013966264, −5.68528982064157406297680762722, −5.36188216881875334932256293820, −4.41902145379602902873521706486, −2.60040665354343198989259265844, −1.99003397666623706140636359617, −0.53721423901972875395860395409, 1.17562641170462494804449119339, 2.16944990770471125199781549451, 3.59536650770341181246324549087, 4.33305436335520724169593539720, 5.17752763925464003748696875925, 6.10191883720432805017355770178, 7.36435819370752465213865380574, 7.75154244813118395385230128057, 8.569088215231780809672440417974, 9.427026461085427104860654013421

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