Properties

Label 2-55-55.2-c1-0-1
Degree $2$
Conductor $55$
Sign $0.521 - 0.853i$
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  + (0.665 + 1.30i)2-s + (−0.130 + 0.822i)3-s + (−0.0875 + 0.120i)4-s + (−1.11 − 1.93i)5-s + (−1.16 + 0.377i)6-s + (−4.16 + 0.659i)7-s + (2.68 + 0.424i)8-s + (2.19 + 0.712i)9-s + (1.78 − 2.74i)10-s + (−0.920 − 3.18i)11-s + (−0.0877 − 0.0877i)12-s + (−0.824 + 0.420i)13-s + (−3.63 − 4.99i)14-s + (1.73 − 0.667i)15-s + (1.32 + 4.06i)16-s + (1.71 + 0.875i)17-s + ⋯
L(s)  = 1  + (0.470 + 0.923i)2-s + (−0.0751 + 0.474i)3-s + (−0.0437 + 0.0602i)4-s + (−0.500 − 0.865i)5-s + (−0.473 + 0.153i)6-s + (−1.57 + 0.249i)7-s + (0.947 + 0.150i)8-s + (0.731 + 0.237i)9-s + (0.564 − 0.869i)10-s + (−0.277 − 0.960i)11-s + (−0.0253 − 0.0253i)12-s + (−0.228 + 0.116i)13-s + (−0.970 − 1.33i)14-s + (0.448 − 0.172i)15-s + (0.330 + 1.01i)16-s + (0.416 + 0.212i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.521 - 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.521 - 0.853i$
Analytic conductor: \(0.439177\)
Root analytic conductor: \(0.662704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :1/2),\ 0.521 - 0.853i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.847753 + 0.475623i\)
\(L(\frac12)\) \(\approx\) \(0.847753 + 0.475623i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.11 + 1.93i)T \)
11 \( 1 + (0.920 + 3.18i)T \)
good2 \( 1 + (-0.665 - 1.30i)T + (-1.17 + 1.61i)T^{2} \)
3 \( 1 + (0.130 - 0.822i)T + (-2.85 - 0.927i)T^{2} \)
7 \( 1 + (4.16 - 0.659i)T + (6.65 - 2.16i)T^{2} \)
13 \( 1 + (0.824 - 0.420i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (-1.71 - 0.875i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (4.39 - 3.19i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.95 + 1.95i)T - 23iT^{2} \)
29 \( 1 + (-0.810 - 0.588i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.131 + 0.403i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.771 + 4.87i)T + (-35.1 + 11.4i)T^{2} \)
41 \( 1 + (-0.339 - 0.467i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (-5.05 - 5.05i)T + 43iT^{2} \)
47 \( 1 + (1.17 + 0.186i)T + (44.6 + 14.5i)T^{2} \)
53 \( 1 + (4.12 + 8.09i)T + (-31.1 + 42.8i)T^{2} \)
59 \( 1 + (-5.47 + 7.53i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-7.40 + 2.40i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + (3.05 + 3.05i)T + 67iT^{2} \)
71 \( 1 + (-2.65 - 8.17i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.854 + 5.39i)T + (-69.4 + 22.5i)T^{2} \)
79 \( 1 + (0.705 - 2.17i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.902 + 1.77i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 - 13.9iT - 89T^{2} \)
97 \( 1 + (2.96 - 1.50i)T + (57.0 - 78.4i)T^{2} \)
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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

−15.88869466989159642641162020774, −14.70078337587029487821289739765, −13.22185345116672715294314000912, −12.59250520005231834801654732154, −10.77042267415301300108535800056, −9.593246161391771818949980476118, −8.130736577956907968351049969863, −6.63552616113035451628106542794, −5.42197055271681493567331948377, −3.94909365557560216297173276784, 2.74666843717127578359086496276, 4.10568300404509122007929224808, 6.73196355452496030552024894922, 7.39813310497359842794180088256, 9.801561978569717485015801067635, 10.60016570277762232262630911746, 12.02294359702003045598072963014, 12.74063914366815483038840671149, 13.52146276927676897866131141808, 15.15922664116966947900724506422

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