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Properties

Label 2-522-261.101-c1-0-28
Degree $2$
Conductor $522$
Sign $-0.696 - 0.717i$
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  + (0.399 − 0.916i)2-s + (−0.432 − 1.67i)3-s + (−0.680 − 0.733i)4-s + (−0.294 − 0.0443i)5-s + (−1.71 − 0.273i)6-s + (−1.01 − 0.937i)7-s + (−0.943 + 0.330i)8-s + (−2.62 + 1.45i)9-s + (−0.158 + 0.251i)10-s + (−3.89 + 3.35i)11-s + (−0.934 + 1.45i)12-s + (1.33 − 1.95i)13-s + (−1.26 + 0.551i)14-s + (0.0529 + 0.512i)15-s + (−0.0747 + 0.997i)16-s + (−4.21 − 4.21i)17-s + ⋯
L(s)  = 1  + (0.282 − 0.648i)2-s + (−0.249 − 0.968i)3-s + (−0.340 − 0.366i)4-s + (−0.131 − 0.0198i)5-s + (−0.698 − 0.111i)6-s + (−0.382 − 0.354i)7-s + (−0.333 + 0.116i)8-s + (−0.875 + 0.484i)9-s + (−0.0500 + 0.0796i)10-s + (−1.17 + 1.01i)11-s + (−0.269 + 0.420i)12-s + (0.370 − 0.543i)13-s + (−0.337 + 0.147i)14-s + (0.0136 + 0.132i)15-s + (−0.0186 + 0.249i)16-s + (−1.02 − 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.696 - 0.717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(522\)    =    \(2 \cdot 3^{2} \cdot 29\)
Sign: $-0.696 - 0.717i$
Analytic conductor: \(4.16819\)
Root analytic conductor: \(2.04161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{522} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 522,\ (\ :1/2),\ -0.696 - 0.717i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.221688 + 0.524091i\)
\(L(\frac12)\) \(\approx\) \(0.221688 + 0.524091i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.399 + 0.916i)T \)
3 \( 1 + (0.432 + 1.67i)T \)
29 \( 1 + (-5.36 - 0.456i)T \)
good5 \( 1 + (0.294 + 0.0443i)T + (4.77 + 1.47i)T^{2} \)
7 \( 1 + (1.01 + 0.937i)T + (0.523 + 6.98i)T^{2} \)
11 \( 1 + (3.89 - 3.35i)T + (1.63 - 10.8i)T^{2} \)
13 \( 1 + (-1.33 + 1.95i)T + (-4.74 - 12.1i)T^{2} \)
17 \( 1 + (4.21 + 4.21i)T + 17iT^{2} \)
19 \( 1 + (3.18 + 2.00i)T + (8.24 + 17.1i)T^{2} \)
23 \( 1 + (-6.76 - 2.65i)T + (16.8 + 15.6i)T^{2} \)
31 \( 1 + (4.52 - 3.34i)T + (9.13 - 29.6i)T^{2} \)
37 \( 1 + (1.84 + 5.28i)T + (-28.9 + 23.0i)T^{2} \)
41 \( 1 + (3.56 + 0.954i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-5.20 + 7.05i)T + (-12.6 - 41.0i)T^{2} \)
47 \( 1 + (-5.09 - 5.91i)T + (-7.00 + 46.4i)T^{2} \)
53 \( 1 + (10.6 + 8.52i)T + (11.7 + 51.6i)T^{2} \)
59 \( 1 + (9.44 - 5.45i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.08 - 0.115i)T + (60.8 - 4.55i)T^{2} \)
67 \( 1 + (-6.64 + 0.497i)T + (66.2 - 9.98i)T^{2} \)
71 \( 1 + (-9.76 + 4.70i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-0.243 + 2.16i)T + (-71.1 - 16.2i)T^{2} \)
79 \( 1 + (-1.35 + 7.14i)T + (-73.5 - 28.8i)T^{2} \)
83 \( 1 + (2.93 - 9.51i)T + (-68.5 - 46.7i)T^{2} \)
89 \( 1 + (4.76 - 0.537i)T + (86.7 - 19.8i)T^{2} \)
97 \( 1 + (8.19 + 15.5i)T + (-54.6 + 80.1i)T^{2} \)
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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

−10.72009013448365248177518604432, −9.530510151510712470310762910631, −8.528166039411321910642519558674, −7.41080257636998442269433603536, −6.78181741323739180103733149223, −5.48725804762670731269901378892, −4.64413671483004176318106021221, −3.06866060190474433206496812836, −2.06467610773164520450190485168, −0.29554793643550900764334582082, 2.82346531186740470942264425880, 3.92390337621552440356813908271, 4.88500019433557632018529663443, 5.93602196547412598135032034598, 6.48925223556754882038236373931, 8.062379918334652187473612167251, 8.681947331532410256107279951958, 9.515799121441746319991538573740, 10.75714694891493173001187940563, 11.07676688383873511102703960707

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