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Properties

Label 2-1950-13.4-c1-0-16
Degree $2$
Conductor $1950$
Sign $-0.252 - 0.967i$
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 + (2.59 − 1.5i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + (−3.23 − 1.86i)11-s − 0.999·12-s + (0.866 + 3.5i)13-s + 3·14-s + (−0.5 + 0.866i)16-s + (2 + 3.46i)17-s − 0.999i·18-s + (−1.96 + 1.13i)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 + (0.981 − 0.566i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.974 − 0.562i)11-s − 0.288·12-s + (0.240 + 0.970i)13-s + 0.801·14-s + (−0.125 + 0.216i)16-s + (0.485 + 0.840i)17-s − 0.235i·18-s + (−0.450 + 0.260i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.252 - 0.967i$
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: \((2,\ 1950,\ (\ :1/2),\ -0.252 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.245098198\)
\(L(\frac12)\) \(\approx\) \(2.245098198\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 3.5i)T \)
good7 \( 1 + (-2.59 + 1.5i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.23 + 1.86i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.96 - 1.13i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.73 - 3i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.73 + 4.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.92iT - 31T^{2} \)
37 \( 1 + (-6.86 - 3.96i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.46 - 2i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3 - 5.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.464iT - 47T^{2} \)
53 \( 1 - 3.73T + 53T^{2} \)
59 \( 1 + (-3.92 + 2.26i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.73 - 6.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.73 + 2.73i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.803 - 0.464i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 - 16.9T + 79T^{2} \)
83 \( 1 - 2.53iT - 83T^{2} \)
89 \( 1 + (8.76 + 5.06i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.7 - 6.19i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.435033476372303358755234199393, −8.211124440196015943543689641001, −8.076572830327416344053093679582, −6.93535933181428478833254541177, −6.06753888335347049250698798736, −5.36852845469605457121441261892, −4.47622025072086611856675238705, −3.95308003957921175125626079283, −2.78220549158276355983459502164, −1.40157107495633824316268577691, 0.71488147486652170105763602358, 2.16982683995393301914996609474, 2.69862954522898376000592264296, 4.11710770944978326839213544625, 5.11725511031364629754326143424, 5.47288094243062781856749572738, 6.41312899975011534619386835607, 7.53841937343450688499038946371, 7.933665477881748934059157640508, 8.898922862861383090803088728260

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