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Properties

Label 2-1950-65.49-c1-0-25
Degree $2$
Conductor $1950$
Sign $0.918 - 0.395i$
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 + (0.866 − 0.5i)3-s + (−0.499 + 0.866i)4-s + (0.866 + 0.499i)6-s + (−1.16 + 2.01i)7-s − 0.999·8-s + (0.499 − 0.866i)9-s + (4.62 − 2.67i)11-s + 0.999i·12-s + (0.161 − 3.60i)13-s − 2.32·14-s + (−0.5 − 0.866i)16-s + (−3.46 − 2i)17-s + 0.999·18-s + (3.48 + 2.01i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.499 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (0.353 + 0.204i)6-s + (−0.438 + 0.760i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (1.39 − 0.805i)11-s + 0.288i·12-s + (0.0447 − 0.998i)13-s − 0.620·14-s + (−0.125 − 0.216i)16-s + (−0.840 − 0.485i)17-s + 0.235·18-s + (0.799 + 0.461i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.395i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.918 - 0.395i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.918 - 0.395i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.561448529\)
\(L(\frac12)\) \(\approx\) \(2.561448529\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
13 \( 1 + (-0.161 + 3.60i)T \)
good7 \( 1 + (1.16 - 2.01i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.62 + 2.67i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.46 + 2i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.48 - 2.01i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.27 + 2.46i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.14 - 3.71i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.47iT - 31T^{2} \)
37 \( 1 + (-1.57 - 2.72i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.29 + 1.32i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-10.6 - 6.12i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 1.81T + 47T^{2} \)
53 \( 1 + 5.48iT - 53T^{2} \)
59 \( 1 + (5.87 + 3.39i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.267 - 0.464i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.05 + 3.55i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-13.7 - 7.93i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 13.5T + 73T^{2} \)
79 \( 1 - 7.96T + 79T^{2} \)
83 \( 1 + 11.3T + 83T^{2} \)
89 \( 1 + (-1.50 + 0.869i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.05 - 13.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.112046801637595731988571125468, −8.463401873705877828923770012143, −7.66020588120964145359971329644, −6.75478031260442869001927116265, −6.15459189822235311228096831073, −5.40751844488913226990472319295, −4.29491036612921320696189053399, −3.30142727630169011062049277272, −2.65374756886063324105447613413, −0.973413954872906515174189438878, 1.15567842493576409921750497186, 2.22135470800740461649845895693, 3.37799697286917854497677101114, 4.19933034420285553133654478880, 4.56994038281822719446993627088, 5.94625196019434457000108352901, 6.91309241680117788588360007956, 7.32156854178403839738463723826, 8.763017054557095285910548391395, 9.265093599991821611013446891314

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