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Properties

Label 2-561-33.32-c1-0-24
Degree $2$
Conductor $561$
Sign $0.711 - 0.702i$
Analytic cond. $4.47960$
Root an. cond. $2.11650$
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  − 2.24·2-s + (1.63 − 0.572i)3-s + 3.01·4-s + 2.28i·5-s + (−3.66 + 1.28i)6-s + 2.48i·7-s − 2.28·8-s + (2.34 − 1.87i)9-s − 5.12i·10-s + (2.99 − 1.42i)11-s + (4.93 − 1.72i)12-s − 2.05i·13-s − 5.57i·14-s + (1.30 + 3.73i)15-s − 0.922·16-s + 17-s + ⋯
L(s)  = 1  − 1.58·2-s + (0.943 − 0.330i)3-s + 1.50·4-s + 1.02i·5-s + (−1.49 + 0.523i)6-s + 0.940i·7-s − 0.807·8-s + (0.781 − 0.623i)9-s − 1.61i·10-s + (0.903 − 0.428i)11-s + (1.42 − 0.498i)12-s − 0.571i·13-s − 1.49i·14-s + (0.337 + 0.964i)15-s − 0.230·16-s + 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 561 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 561 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(561\)    =    \(3 \cdot 11 \cdot 17\)
Sign: $0.711 - 0.702i$
Analytic conductor: \(4.47960\)
Root analytic conductor: \(2.11650\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{561} (494, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 561,\ (\ :1/2),\ 0.711 - 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.917312 + 0.376770i\)
\(L(\frac12)\) \(\approx\) \(0.917312 + 0.376770i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.63 + 0.572i)T \)
11 \( 1 + (-2.99 + 1.42i)T \)
17 \( 1 - T \)
good2 \( 1 + 2.24T + 2T^{2} \)
5 \( 1 - 2.28iT - 5T^{2} \)
7 \( 1 - 2.48iT - 7T^{2} \)
13 \( 1 + 2.05iT - 13T^{2} \)
19 \( 1 - 0.765iT - 19T^{2} \)
23 \( 1 - 9.13iT - 23T^{2} \)
29 \( 1 + 1.59T + 29T^{2} \)
31 \( 1 + 5.20T + 31T^{2} \)
37 \( 1 + 0.670T + 37T^{2} \)
41 \( 1 - 4.72T + 41T^{2} \)
43 \( 1 - 4.60iT - 43T^{2} \)
47 \( 1 + 2.41iT - 47T^{2} \)
53 \( 1 - 6.66iT - 53T^{2} \)
59 \( 1 + 5.60iT - 59T^{2} \)
61 \( 1 - 7.57iT - 61T^{2} \)
67 \( 1 - 7.19T + 67T^{2} \)
71 \( 1 + 6.76iT - 71T^{2} \)
73 \( 1 + 4.71iT - 73T^{2} \)
79 \( 1 - 13.3iT - 79T^{2} \)
83 \( 1 - 9.60T + 83T^{2} \)
89 \( 1 + 1.01iT - 89T^{2} \)
97 \( 1 + 9.54T + 97T^{2} \)
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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

−10.66064718550696946175086592521, −9.558586218752108275970264739962, −9.241272372981083566222860634354, −8.296385025682219929375438184510, −7.55504722886773372337133403395, −6.83076626619948736224994863281, −5.79231492973960883258212974945, −3.61514230043418787827443549402, −2.64032335206207703182939712835, −1.47424764666854099728174148046, 0.976980530352380587202030149373, 2.12872911574080252203092315184, 3.93890612947298075088388738571, 4.73133185974218544544979706736, 6.69765512611942117638441850355, 7.39514021934042740151719516825, 8.339019684403159721534320416090, 8.946560950887423969313132420372, 9.500232388837340899435257995278, 10.31139227160420283072249623417

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