↑

Properties

Label 2-975-195.44-c1-0-10
Degree $2$
Conductor $975$
Sign $-0.999 + 0.0367i$
Analytic cond. $7.78541$
Root an. cond. $2.79023$
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  + (0.483 − 0.483i)2-s + (−0.793 + 1.53i)3-s + 1.53i·4-s + (0.361 + 1.12i)6-s + (−1.24 + 1.24i)7-s + (1.70 + 1.70i)8-s + (−1.74 − 2.44i)9-s + (0.387 − 0.387i)11-s + (−2.35 − 1.21i)12-s + (−3.49 − 0.874i)13-s + 1.20i·14-s − 1.41·16-s + 6.75i·17-s + (−2.02 − 0.339i)18-s + (−1.33 + 1.33i)19-s + ⋯
L(s)  = 1  + (0.342 − 0.342i)2-s + (−0.458 + 0.888i)3-s + 0.765i·4-s + (0.147 + 0.460i)6-s + (−0.470 + 0.470i)7-s + (0.604 + 0.604i)8-s + (−0.580 − 0.814i)9-s + (0.116 − 0.116i)11-s + (−0.680 − 0.350i)12-s + (−0.970 − 0.242i)13-s + 0.322i·14-s − 0.352·16-s + 1.63i·17-s + (−0.477 − 0.0800i)18-s + (−0.305 + 0.305i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0367i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0367i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.999 + 0.0367i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (824, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.999 + 0.0367i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0136692 - 0.744142i\)
\(L(\frac12)\) \(\approx\) \(0.0136692 - 0.744142i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.793 - 1.53i)T \)
5 \( 1 \)
13 \( 1 + (3.49 + 0.874i)T \)
good2 \( 1 + (-0.483 + 0.483i)T - 2iT^{2} \)
7 \( 1 + (1.24 - 1.24i)T - 7iT^{2} \)
11 \( 1 + (-0.387 + 0.387i)T - 11iT^{2} \)
17 \( 1 - 6.75iT - 17T^{2} \)
19 \( 1 + (1.33 - 1.33i)T - 19iT^{2} \)
23 \( 1 + 1.69iT - 23T^{2} \)
29 \( 1 + 3.85iT - 29T^{2} \)
31 \( 1 + (-4.06 + 4.06i)T - 31iT^{2} \)
37 \( 1 + (2.36 - 2.36i)T - 37iT^{2} \)
41 \( 1 + (5.72 + 5.72i)T + 41iT^{2} \)
43 \( 1 + 11.7T + 43T^{2} \)
47 \( 1 + (5.99 + 5.99i)T + 47iT^{2} \)
53 \( 1 - 8.81T + 53T^{2} \)
59 \( 1 + (-1.30 + 1.30i)T - 59iT^{2} \)
61 \( 1 + 1.16T + 61T^{2} \)
67 \( 1 + (-4.66 - 4.66i)T + 67iT^{2} \)
71 \( 1 + (-0.915 - 0.915i)T + 71iT^{2} \)
73 \( 1 + (8.11 - 8.11i)T - 73iT^{2} \)
79 \( 1 - 3.85T + 79T^{2} \)
83 \( 1 + (11.9 - 11.9i)T - 83iT^{2} \)
89 \( 1 + (5.99 - 5.99i)T - 89iT^{2} \)
97 \( 1 + (1.03 + 1.03i)T + 97iT^{2} \)
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

−10.28667998361687605101295874053, −9.929471707412588966710353163641, −8.659746476626408493206771902011, −8.240349435784642520147122968571, −6.89899504445222406011586034161, −5.99343176511704680192530175754, −5.04210782952964906884593033025, −4.10937576119593844274432661131, −3.37734456724736667533913132597, −2.27068794839462763833604836323, 0.31113684450754993330915774184, 1.67747153063380941330207890161, 3.02997189213526594185967840403, 4.74204799930398580255882706428, 5.15126432624555072160763597264, 6.34039290687281436067174427565, 6.96449302972920609775757992617, 7.40666524961540687012896692385, 8.731303516542057365081083536394, 9.784380822886616016488938944491

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