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Properties

Label 2-55-11.9-c1-0-2
Degree $2$
Conductor $55$
Sign $0.977 + 0.209i$
Analytic cond. $0.439177$
Root an. cond. $0.662704$
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  + (0.147 − 0.453i)2-s + (−0.261 + 0.189i)3-s + (1.43 + 1.04i)4-s + (−0.309 − 0.951i)5-s + (0.0476 + 0.146i)6-s + (−2.17 − 1.57i)7-s + (1.45 − 1.05i)8-s + (−0.894 + 2.75i)9-s − 0.477·10-s + (−2.79 − 1.79i)11-s − 0.572·12-s + (−1.44 + 4.43i)13-s + (−1.03 + 0.753i)14-s + (0.261 + 0.189i)15-s + (0.829 + 2.55i)16-s + (−1.42 − 4.39i)17-s + ⋯
L(s)  = 1  + (0.104 − 0.320i)2-s + (−0.150 + 0.109i)3-s + (0.716 + 0.520i)4-s + (−0.138 − 0.425i)5-s + (0.0194 + 0.0598i)6-s + (−0.821 − 0.596i)7-s + (0.514 − 0.374i)8-s + (−0.298 + 0.917i)9-s − 0.150·10-s + (−0.841 − 0.540i)11-s − 0.165·12-s + (−0.400 + 1.23i)13-s + (−0.277 + 0.201i)14-s + (0.0674 + 0.0490i)15-s + (0.207 + 0.638i)16-s + (−0.346 − 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 + 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.977 + 0.209i$
Analytic conductor: \(0.439177\)
Root analytic conductor: \(0.662704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :1/2),\ 0.977 + 0.209i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.888300 - 0.0941154i\)
\(L(\frac12)\) \(\approx\) \(0.888300 - 0.0941154i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (2.79 + 1.79i)T \)
good2 \( 1 + (-0.147 + 0.453i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (0.261 - 0.189i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + (2.17 + 1.57i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (1.44 - 4.43i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.42 + 4.39i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-3.51 + 2.55i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 2.77T + 23T^{2} \)
29 \( 1 + (-2.43 - 1.77i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.737 + 2.26i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-8.61 - 6.25i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.78 + 1.29i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 7.06T + 43T^{2} \)
47 \( 1 + (3.52 - 2.56i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.95 - 6.02i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (9.50 + 6.90i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (1.23 + 3.78i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 7.31T + 67T^{2} \)
71 \( 1 + (-0.369 - 1.13i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.826 + 0.600i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.08 + 3.33i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-3.43 - 10.5i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 2.76T + 89T^{2} \)
97 \( 1 + (-5.72 + 17.6i)T + (-78.4 - 57.0i)T^{2} \)
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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

−15.75056155440062040278034266744, −13.77825355802615383516846051634, −13.10901603888172762826625632825, −11.72468626707099500079624711830, −10.94055486123167482789326732604, −9.566302831696967268366743321529, −7.896252055725894899091818898829, −6.75492738464691154106956630459, −4.77179832299571594616882858543, −2.86878193548490114280889880405, 2.92168491018701487635908797969, 5.55342796449956502906124677934, 6.52183756843724344126252072339, 7.85046637762129686202667974854, 9.708002093866407336057764557194, 10.68483016970310806188454072607, 12.04389045753621286005671996665, 12.99109177053680068805776304200, 14.76110512962494513799275434135, 15.24017275597784785293328193792

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