↑

Properties

Label 2-952-952.781-c1-0-91
Degree $2$
Conductor $952$
Sign $0.622 + 0.782i$
Analytic cond. $7.60175$
Root an. cond. $2.75712$
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  + (−1.37 + 0.311i)2-s + (1.23 − 2.14i)3-s + (1.80 − 0.859i)4-s + (0.751 + 1.30i)5-s + (−1.03 + 3.33i)6-s + (2.55 − 0.683i)7-s + (−2.22 + 1.74i)8-s + (−1.55 − 2.69i)9-s + (−1.44 − 1.56i)10-s + (1.94 − 3.36i)11-s + (0.391 − 4.92i)12-s + 6.06i·13-s + (−3.31 + 1.73i)14-s + 3.71·15-s + (2.52 − 3.10i)16-s + (−4.04 − 0.785i)17-s + ⋯
L(s)  = 1  + (−0.975 + 0.220i)2-s + (0.713 − 1.23i)3-s + (0.902 − 0.429i)4-s + (0.336 + 0.582i)5-s + (−0.423 + 1.36i)6-s + (0.966 − 0.258i)7-s + (−0.786 + 0.618i)8-s + (−0.517 − 0.897i)9-s + (−0.456 − 0.493i)10-s + (0.585 − 1.01i)11-s + (0.113 − 1.42i)12-s + 1.68i·13-s + (−0.885 + 0.464i)14-s + 0.959·15-s + (0.630 − 0.776i)16-s + (−0.981 − 0.190i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.622 + 0.782i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(952\)    =    \(2^{3} \cdot 7 \cdot 17\)
Sign: $0.622 + 0.782i$
Analytic conductor: \(7.60175\)
Root analytic conductor: \(2.75712\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{952} (781, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 952,\ (\ :1/2),\ 0.622 + 0.782i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45792 - 0.702679i\)
\(L(\frac12)\) \(\approx\) \(1.45792 - 0.702679i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 - 0.311i)T \)
7 \( 1 + (-2.55 + 0.683i)T \)
17 \( 1 + (4.04 + 0.785i)T \)
good3 \( 1 + (-1.23 + 2.14i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.751 - 1.30i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.94 + 3.36i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 6.06iT - 13T^{2} \)
19 \( 1 + (-1.76 + 1.01i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.71 + 3.30i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.640T + 29T^{2} \)
31 \( 1 + (2.61 + 1.50i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.52 - 7.83i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.20iT - 41T^{2} \)
43 \( 1 - 5.70iT - 43T^{2} \)
47 \( 1 + (4.60 + 7.98i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.03 + 1.75i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.139 + 0.0805i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.30 - 5.72i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.20 + 3.00i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.95iT - 71T^{2} \)
73 \( 1 + (7.57 + 4.37i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.11 - 0.642i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.13iT - 83T^{2} \)
89 \( 1 + (-2.43 - 4.22i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.2iT - 97T^{2} \)
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

−9.625107392228627254648580890216, −8.754347686146294857782106748666, −8.410200343773562002486250785491, −7.34861803497204107229593724569, −6.74877510579488929727595075852, −6.27392142589327176870548616327, −4.66806808371906779962732950429, −2.95530969606275276618284047888, −2.03671719828283726876947194129, −1.14337666708429430942052609898, 1.40141486364166347746446141272, 2.63274474358406959792345333838, 3.72883088206447308921631272841, 4.81199827589294179649969382964, 5.63340963613399248971430554202, 7.18480516000665947215846636888, 7.948983841877930176658958885695, 8.903491191258144742974715406338, 9.161415947344153457549543647307, 9.976424097759839061216511847858

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