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Properties

Label 2-527-31.4-c1-0-29
Degree $2$
Conductor $527$
Sign $0.971 - 0.238i$
Analytic cond. $4.20811$
Root an. cond. $2.05136$
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  + (0.0573 + 0.176i)2-s + (−0.417 + 1.28i)3-s + (1.59 − 1.15i)4-s + 2.24·5-s − 0.250·6-s + (2.41 − 1.75i)7-s + (0.595 + 0.432i)8-s + (0.953 + 0.692i)9-s + (0.128 + 0.396i)10-s + (−4.52 + 3.28i)11-s + (0.819 + 2.52i)12-s + (1.77 − 5.45i)13-s + (0.449 + 0.326i)14-s + (−0.936 + 2.88i)15-s + (1.17 − 3.60i)16-s + (−0.809 − 0.587i)17-s + ⋯
L(s)  = 1  + (0.0405 + 0.124i)2-s + (−0.240 + 0.741i)3-s + (0.795 − 0.577i)4-s + 1.00·5-s − 0.102·6-s + (0.914 − 0.664i)7-s + (0.210 + 0.152i)8-s + (0.317 + 0.230i)9-s + (0.0407 + 0.125i)10-s + (−1.36 + 0.990i)11-s + (0.236 + 0.728i)12-s + (0.491 − 1.51i)13-s + (0.120 + 0.0872i)14-s + (−0.241 + 0.744i)15-s + (0.293 − 0.902i)16-s + (−0.196 − 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.238i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 527 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 - 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(527\)    =    \(17 \cdot 31\)
Sign: $0.971 - 0.238i$
Analytic conductor: \(4.20811\)
Root analytic conductor: \(2.05136\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{527} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 527,\ (\ :1/2),\ 0.971 - 0.238i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02427 + 0.244587i\)
\(L(\frac12)\) \(\approx\) \(2.02427 + 0.244587i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (5.56 + 0.200i)T \)
good2 \( 1 + (-0.0573 - 0.176i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (0.417 - 1.28i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 - 2.24T + 5T^{2} \)
7 \( 1 + (-2.41 + 1.75i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (4.52 - 3.28i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-1.77 + 5.45i)T + (-10.5 - 7.64i)T^{2} \)
19 \( 1 + (0.737 + 2.27i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (1.65 + 1.20i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.74 - 5.35i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 - 4.55T + 37T^{2} \)
41 \( 1 + (-1.55 - 4.79i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-2.97 - 9.16i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (2.06 - 6.34i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.30 + 1.67i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.552 + 1.69i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 - 3.77T + 61T^{2} \)
67 \( 1 - 5.18T + 67T^{2} \)
71 \( 1 + (12.5 + 9.10i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (10.5 - 7.63i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.758 + 0.551i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.29 + 3.97i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-11.6 + 8.49i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (0.642 - 0.466i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.66433293049378812228480250907, −10.26906427279673849500874999570, −9.546668299270770460370700444927, −7.929057160672936978820432484227, −7.37941728036486641987435688524, −6.05145968071330166627094466768, −5.20415307512504645897847233671, −4.62875298595070656663145961613, −2.72906754982018369325501236581, −1.54239977956998992754757292730, 1.75508451378149840373344474739, 2.32090016091787389824411956478, 3.97695940738487493900274656401, 5.60645751977636580138983623971, 6.14129762848257550075387373534, 7.17113763050249977280918737428, 8.068662689596221231073143683930, 8.864427958006547403103007767828, 10.09750036258226685552818443205, 11.07190489420973985892743584557

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