↑

Properties

Label 2-1950-65.8-c1-0-13
Degree $2$
Conductor $1950$
Sign $0.760 - 0.649i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + (−0.707 − 0.707i)3-s + 4-s + (0.707 + 0.707i)6-s + 4.10i·7-s − 8-s + 1.00i·9-s + (−0.222 + 0.222i)11-s + (−0.707 − 0.707i)12-s + (2.75 + 2.32i)13-s − 4.10i·14-s + 16-s + (−4.74 − 4.74i)17-s − 1.00i·18-s + (5.81 − 5.81i)19-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.408 − 0.408i)3-s + 0.5·4-s + (0.288 + 0.288i)6-s + 1.55i·7-s − 0.353·8-s + 0.333i·9-s + (−0.0670 + 0.0670i)11-s + (−0.204 − 0.204i)12-s + (0.763 + 0.645i)13-s − 1.09i·14-s + 0.250·16-s + (−1.14 − 1.14i)17-s − 0.235i·18-s + (1.33 − 1.33i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.760 - 0.649i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (1243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.760 - 0.649i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.022591096\)
\(L(\frac12)\) \(\approx\) \(1.022591096\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 \)
13 \( 1 + (-2.75 - 2.32i)T \)
good7 \( 1 - 4.10iT - 7T^{2} \)
11 \( 1 + (0.222 - 0.222i)T - 11iT^{2} \)
17 \( 1 + (4.74 + 4.74i)T + 17iT^{2} \)
19 \( 1 + (-5.81 + 5.81i)T - 19iT^{2} \)
23 \( 1 + (-3.55 + 3.55i)T - 23iT^{2} \)
29 \( 1 + 3.67iT - 29T^{2} \)
31 \( 1 + (1.19 + 1.19i)T + 31iT^{2} \)
37 \( 1 - 6.24iT - 37T^{2} \)
41 \( 1 + (-5.97 - 5.97i)T + 41iT^{2} \)
43 \( 1 + (6.75 - 6.75i)T - 43iT^{2} \)
47 \( 1 - 7.67iT - 47T^{2} \)
53 \( 1 + (-3.92 - 3.92i)T + 53iT^{2} \)
59 \( 1 + (4.75 + 4.75i)T + 59iT^{2} \)
61 \( 1 - 14.0T + 61T^{2} \)
67 \( 1 - 5.51T + 67T^{2} \)
71 \( 1 + (-5.57 - 5.57i)T + 71iT^{2} \)
73 \( 1 + 3.88T + 73T^{2} \)
79 \( 1 - 1.49iT - 79T^{2} \)
83 \( 1 - 9.61iT - 83T^{2} \)
89 \( 1 + (-3.26 - 3.26i)T + 89iT^{2} \)
97 \( 1 - 12.1T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.233740257628086062002603812148, −8.640117450171090950350412550943, −7.80344328599452391152868335710, −6.76648794391283072221117898278, −6.39576942508408291649892922724, −5.33879976415811363658243306495, −4.64797625882162887466793990404, −2.89868645196106433532813338271, −2.33970880500205408328941269911, −0.966929438396549968874218323471, 0.64955703393341033366623556902, 1.69980276228494195310731770844, 3.55444321563349688378207761386, 3.81027470705430703945245366511, 5.18843922049198035168526514013, 5.95785260803700090930274273489, 6.95573354229169572811442857515, 7.48000532555577137652450834716, 8.382078726546448491570251588648, 9.100353636375475211918059482597

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