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Properties

Label 2-1950-13.10-c1-0-39
Degree $2$
Conductor $1950$
Sign $-0.0838 + 0.996i$
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 + (−1.42 − 0.823i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (2.08 − 1.20i)11-s + 0.999·12-s + (0.256 − 3.59i)13-s − 1.64·14-s + (−0.5 − 0.866i)16-s + (0.121 − 0.210i)17-s + 0.999i·18-s + (−3.82 − 2.20i)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 + (−0.538 − 0.311i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.628 − 0.362i)11-s + 0.288·12-s + (0.0710 − 0.997i)13-s − 0.439·14-s + (−0.125 − 0.216i)16-s + (0.0295 − 0.0511i)17-s + 0.235i·18-s + (−0.877 − 0.506i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0838 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0838 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.0838 + 0.996i$
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: \((2,\ 1950,\ (\ :1/2),\ -0.0838 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.203609368\)
\(L(\frac12)\) \(\approx\) \(2.203609368\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.256 + 3.59i)T \)
good7 \( 1 + (1.42 + 0.823i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.08 + 1.20i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.121 + 0.210i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.82 + 2.20i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.31 + 7.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.0221 + 0.0383i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.24iT - 31T^{2} \)
37 \( 1 + (-7.74 + 4.47i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.210 + 0.121i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.36 + 5.82i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.29iT - 47T^{2} \)
53 \( 1 - 2.44T + 53T^{2} \)
59 \( 1 + (-8.35 - 4.82i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.31 + 2.27i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.62 + 0.937i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.53 + 3.77i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.70iT - 73T^{2} \)
79 \( 1 + 6.79T + 79T^{2} \)
83 \( 1 + 17.4iT - 83T^{2} \)
89 \( 1 + (-8.69 + 5.02i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (14.3 + 8.25i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.995429671395601098631620835753, −8.387747506462120042882560257640, −7.33041001126799649215482379792, −6.37799149658156709313960236587, −5.79127595234637300816675901730, −4.65261548196734340491981205556, −4.00472838892252586462832455415, −3.14767602460160855910989710422, −2.27843199458267030576814715849, −0.60496654196121165077009889776, 1.60753817027493438474666197685, 2.55545459716081899198211376740, 3.75403778042140959078721640121, 4.30045622450217711422464825113, 5.60545228768200989669664362633, 6.28675307849634976605244008096, 6.86605424290499503365748570578, 7.74066261463041992177576431227, 8.447717708231465394295227264229, 9.407799841304566327340207012666

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