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Properties

Label 2-55-55.9-c1-0-1
Degree $2$
Conductor $55$
Sign $0.908 - 0.416i$
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  + (0.763 + 0.248i)2-s + (1.03 + 1.42i)3-s + (−1.09 − 0.796i)4-s + (−2.16 − 0.550i)5-s + (0.436 + 1.34i)6-s + (0.348 − 0.479i)7-s + (−1.58 − 2.17i)8-s + (−0.0309 + 0.0953i)9-s + (−1.51 − 0.958i)10-s + (1.96 + 2.67i)11-s − 2.38i·12-s + (1.70 + 0.554i)13-s + (0.384 − 0.279i)14-s + (−1.45 − 3.65i)15-s + (0.169 + 0.522i)16-s + (−6.73 + 2.18i)17-s + ⋯
L(s)  = 1  + (0.539 + 0.175i)2-s + (0.597 + 0.822i)3-s + (−0.548 − 0.398i)4-s + (−0.969 − 0.246i)5-s + (0.178 + 0.548i)6-s + (0.131 − 0.181i)7-s + (−0.559 − 0.770i)8-s + (−0.0103 + 0.0317i)9-s + (−0.479 − 0.302i)10-s + (0.591 + 0.806i)11-s − 0.689i·12-s + (0.473 + 0.153i)13-s + (0.102 − 0.0746i)14-s + (−0.376 − 0.944i)15-s + (0.0424 + 0.130i)16-s + (−1.63 + 0.530i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 - 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.908 - 0.416i$
Analytic conductor: \(0.439177\)
Root analytic conductor: \(0.662704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :1/2),\ 0.908 - 0.416i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.996919 + 0.217663i\)
\(L(\frac12)\) \(\approx\) \(0.996919 + 0.217663i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.16 + 0.550i)T \)
11 \( 1 + (-1.96 - 2.67i)T \)
good2 \( 1 + (-0.763 - 0.248i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (-1.03 - 1.42i)T + (-0.927 + 2.85i)T^{2} \)
7 \( 1 + (-0.348 + 0.479i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-1.70 - 0.554i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (6.73 - 2.18i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.85 - 1.34i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 1.49iT - 23T^{2} \)
29 \( 1 + (-2.89 - 2.10i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.90 + 5.86i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-4.31 + 5.93i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (6.80 - 4.94i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 9.51iT - 43T^{2} \)
47 \( 1 + (1.13 + 1.56i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.26 + 0.736i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (0.0309 + 0.0224i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (1.06 + 3.27i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 6.79iT - 67T^{2} \)
71 \( 1 + (3.64 + 11.2i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (4.01 - 5.51i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.39 - 4.30i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-5.65 + 1.83i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 - 6.21T + 89T^{2} \)
97 \( 1 + (5.11 + 1.66i)T + (78.4 + 57.0i)T^{2} \)
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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

−15.16239817916370456749073085468, −14.65224418265346909963217175150, −13.33765135096727590000641620308, −12.19870200888880488839064415212, −10.68830640646073436465651718701, −9.364119321420619534532872188352, −8.506518750202541310893834288021, −6.60656087116736989004382991985, −4.51009635331319408948221638034, −3.94035739675604629275813183242, 3.00400563869808937727519263009, 4.53408775954562020351447476323, 6.71260838965373709044662413602, 8.178302155326042387635419502745, 8.791261974689878466960631957649, 11.11009599709416320270030758813, 12.02033962773150856970228683767, 13.23059414419176313272175462810, 13.82461743971582872662047555216, 14.97256549396788832510943931702

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