Properties

Label 2-1950-13.10-c1-0-2
Degree $2$
Conductor $1950$
Sign $-0.881 - 0.472i$
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 + (−1.14 − 0.661i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (−3.99 + 2.30i)11-s + 0.999·12-s + (−3.20 − 1.66i)13-s − 1.32·14-s + (−0.5 − 0.866i)16-s + (−2 + 3.46i)17-s + 0.999i·18-s + (1.98 + 1.14i)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 + (−0.432 − 0.249i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (−1.20 + 0.695i)11-s + 0.288·12-s + (−0.887 − 0.460i)13-s − 0.353·14-s + (−0.125 − 0.216i)16-s + (−0.485 + 0.840i)17-s + 0.235i·18-s + (0.455 + 0.262i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.881 - 0.472i$
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: \((2,\ 1950,\ (\ :1/2),\ -0.881 - 0.472i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4152215841\)
\(L(\frac12)\) \(\approx\) \(0.4152215841\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.20 + 1.66i)T \)
good7 \( 1 + (1.14 + 0.661i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.99 - 2.30i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.98 - 1.14i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.33 + 7.50i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.01 - 1.75i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 10.1iT - 31T^{2} \)
37 \( 1 + (5.89 - 3.40i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.02 - 2.32i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.30 - 7.45i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 9.10iT - 47T^{2} \)
53 \( 1 + 0.826T + 53T^{2} \)
59 \( 1 + (-2.72 - 1.57i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.267 - 0.464i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.75 + 1.59i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (9.81 + 5.66i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.28iT - 73T^{2} \)
79 \( 1 - 2.96T + 79T^{2} \)
83 \( 1 - 15.8iT - 83T^{2} \)
89 \( 1 + (-10.2 + 5.93i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.63 - 4.40i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.952793208184251584616066397977, −8.703182210766958754349080875953, −8.037046625395043654548005041291, −7.05822561820221567114689031689, −6.30741088809851691900515055376, −5.06529747056633343416298046553, −4.80617588849071178934974944818, −3.62993990458785228971617107922, −2.85560236302605409194866738618, −1.91621691774840402290996696314, 0.10229906654547800742281297111, 2.11540403919705122709696245699, 2.85541899615371069870083059123, 3.79172904180426391271724037433, 4.97590445252595667676136307475, 5.64267835502277655130218157146, 6.42704774794303005452209506500, 7.51205791680018609590008579258, 7.64211927146379153467014128237, 8.788500982258015288748054066304

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