Properties

Label 2-522-261.115-c1-0-7
Degree $2$
Conductor $522$
Sign $-0.204 - 0.978i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.866 + 1.5i)3-s + (0.499 + 0.866i)4-s + (1 + 1.73i)5-s − 1.73i·6-s + (−0.5 + 0.866i)7-s − 0.999i·8-s + (−1.5 + 2.59i)9-s − 1.99i·10-s + (−0.866 − 0.5i)11-s + (−0.866 + 1.49i)12-s + (2 + 3.46i)13-s + (0.866 − 0.499i)14-s + (−1.73 + 3i)15-s + (−0.5 + 0.866i)16-s − 2i·17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.499 + 0.866i)3-s + (0.249 + 0.433i)4-s + (0.447 + 0.774i)5-s − 0.707i·6-s + (−0.188 + 0.327i)7-s − 0.353i·8-s + (−0.5 + 0.866i)9-s − 0.632i·10-s + (−0.261 − 0.150i)11-s + (−0.250 + 0.433i)12-s + (0.554 + 0.960i)13-s + (0.231 − 0.133i)14-s + (−0.447 + 0.774i)15-s + (−0.125 + 0.216i)16-s − 0.485i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(522\)    =    \(2 \cdot 3^{2} \cdot 29\)
Sign: $-0.204 - 0.978i$
Analytic conductor: \(4.16819\)
Root analytic conductor: \(2.04161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{522} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 522,\ (\ :1/2),\ -0.204 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.760203 + 0.935453i\)
\(L(\frac12)\) \(\approx\) \(0.760203 + 0.935453i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 1.5i)T \)
29 \( 1 + (5.33 + 0.767i)T \)
good5 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2 - 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - iT - 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
31 \( 1 + (3.46 - 2i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + (-1.73 + i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.79 - 4.5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.19 + 3i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-11.2 - 6.5i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + (-3.46 - 2i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2 + 3.46i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 2iT - 89T^{2} \)
97 \( 1 + (-6.92 - 4i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.95985614923408925976881477567, −10.14821961862398371046458894333, −9.397745291342544600148839526040, −8.781997697453234764718240991087, −7.73117960773942752842724811904, −6.65054285836619692060347953310, −5.55683410122019095833534661071, −4.14393261632435862506790836279, −3.07717395764334057481192608604, −2.11094735221923997629919290130, 0.823847506012596813373372613877, 2.10047390762500387938419578849, 3.61512115301385519379151004856, 5.32709319370395485649534850855, 6.10761774798077706922814241669, 7.21630099609704423589416707605, 7.941316809998093987961059294570, 8.787668818025936976181157198996, 9.427078463489441668130903620586, 10.46561587941067090742181261663

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