↑

Properties

Label 2-522-261.103-c1-0-12
Degree $2$
Conductor $522$
Sign $0.996 - 0.0797i$
Analytic cond. $4.16819$
Root an. cond. $2.04161$
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  + (−0.0747 − 0.997i)2-s + (1.36 + 1.06i)3-s + (−0.988 + 0.149i)4-s + (0.159 − 0.108i)5-s + (0.961 − 1.44i)6-s + (−0.0252 − 0.00380i)7-s + (0.222 + 0.974i)8-s + (0.726 + 2.91i)9-s + (−0.120 − 0.151i)10-s + (−0.978 + 0.301i)11-s + (−1.50 − 0.850i)12-s + (4.49 + 4.17i)13-s + (−0.00190 + 0.0254i)14-s + (0.333 + 0.0216i)15-s + (0.955 − 0.294i)16-s + 6.74·17-s + ⋯
L(s)  = 1  + (−0.0528 − 0.705i)2-s + (0.788 + 0.615i)3-s + (−0.494 + 0.0745i)4-s + (0.0713 − 0.0486i)5-s + (0.392 − 0.588i)6-s + (−0.00954 − 0.00143i)7-s + (0.0786 + 0.344i)8-s + (0.242 + 0.970i)9-s + (−0.0380 − 0.0477i)10-s + (−0.295 + 0.0910i)11-s + (−0.435 − 0.245i)12-s + (1.24 + 1.15i)13-s + (−0.000509 + 0.00680i)14-s + (0.0862 + 0.00558i)15-s + (0.238 − 0.0736i)16-s + 1.63·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(522\)    =    \(2 \cdot 3^{2} \cdot 29\)
Sign: $0.996 - 0.0797i$
Analytic conductor: \(4.16819\)
Root analytic conductor: \(2.04161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{522} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 522,\ (\ :1/2),\ 0.996 - 0.0797i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80403 + 0.0720812i\)
\(L(\frac12)\) \(\approx\) \(1.80403 + 0.0720812i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0747 + 0.997i)T \)
3 \( 1 + (-1.36 - 1.06i)T \)
29 \( 1 + (5.31 - 0.876i)T \)
good5 \( 1 + (-0.159 + 0.108i)T + (1.82 - 4.65i)T^{2} \)
7 \( 1 + (0.0252 + 0.00380i)T + (6.68 + 2.06i)T^{2} \)
11 \( 1 + (0.978 - 0.301i)T + (9.08 - 6.19i)T^{2} \)
13 \( 1 + (-4.49 - 4.17i)T + (0.971 + 12.9i)T^{2} \)
17 \( 1 - 6.74T + 17T^{2} \)
19 \( 1 + (2.37 + 2.98i)T + (-4.22 + 18.5i)T^{2} \)
23 \( 1 + (-0.277 + 3.70i)T + (-22.7 - 3.42i)T^{2} \)
31 \( 1 + (-6.09 + 4.15i)T + (11.3 - 28.8i)T^{2} \)
37 \( 1 + (-1.44 - 6.31i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 + (-3.65 + 6.32i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-8.51 - 5.80i)T + (15.7 + 40.0i)T^{2} \)
47 \( 1 + (4.35 - 1.34i)T + (38.8 - 26.4i)T^{2} \)
53 \( 1 + (5.82 + 2.80i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + (-6.02 + 10.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (9.74 + 1.46i)T + (58.2 + 17.9i)T^{2} \)
67 \( 1 + (4.45 + 1.37i)T + (55.3 + 37.7i)T^{2} \)
71 \( 1 + (-3.12 + 13.7i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (8.72 - 4.20i)T + (45.5 - 57.0i)T^{2} \)
79 \( 1 + (6.43 - 5.97i)T + (5.90 - 78.7i)T^{2} \)
83 \( 1 + (0.367 - 0.936i)T + (-60.8 - 56.4i)T^{2} \)
89 \( 1 + (-5.49 - 2.64i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + (-6.18 + 15.7i)T + (-71.1 - 65.9i)T^{2} \)
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

−10.90445594871404310721012128587, −9.869442964387242060388406866385, −9.301285889442868131760030747936, −8.432649413233867371960667678059, −7.61861449981220022542220132324, −6.16060720370233405395623109629, −4.86728307528570676605933146271, −3.93884871895334120247087602946, −2.98681805493503028306088250771, −1.65673812154649475404225958583, 1.20305294772277280462142555388, 3.00069287067999847771239458605, 3.96403607427812802591235648769, 5.66973822740405434222929675043, 6.18165650086214339019372889218, 7.58636962057066963502504476685, 7.950288872592823581441839188778, 8.792743720549549786100117037875, 9.822297949951962528989187075776, 10.60813345467551976802453492964

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