Properties

Label 2-55-55.14-c1-0-0
Degree $2$
Conductor $55$
Sign $0.0986 - 0.995i$
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  + (1.18 + 1.63i)2-s + (−2.49 + 0.809i)3-s + (−0.647 + 1.99i)4-s + (1.54 − 1.61i)5-s + (−4.29 − 3.11i)6-s + (0.918 + 0.298i)7-s + (−0.183 + 0.0595i)8-s + (3.12 − 2.27i)9-s + (4.48 + 0.596i)10-s + (−3.31 + 0.189i)11-s − 5.49i·12-s + (−2.65 − 3.66i)13-s + (0.603 + 1.85i)14-s + (−2.53 + 5.28i)15-s + (3.07 + 2.23i)16-s + (−1.96 + 2.69i)17-s + ⋯
L(s)  = 1  + (0.841 + 1.15i)2-s + (−1.43 + 0.467i)3-s + (−0.323 + 0.996i)4-s + (0.689 − 0.724i)5-s + (−1.75 − 1.27i)6-s + (0.347 + 0.112i)7-s + (−0.0648 + 0.0210i)8-s + (1.04 − 0.757i)9-s + (1.41 + 0.188i)10-s + (−0.998 + 0.0572i)11-s − 1.58i·12-s + (−0.737 − 1.01i)13-s + (0.161 + 0.496i)14-s + (−0.653 + 1.36i)15-s + (0.768 + 0.558i)16-s + (−0.475 + 0.654i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0986 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0986 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.0986 - 0.995i$
Analytic conductor: \(0.439177\)
Root analytic conductor: \(0.662704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :1/2),\ 0.0986 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.693327 + 0.628010i\)
\(L(\frac12)\) \(\approx\) \(0.693327 + 0.628010i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.54 + 1.61i)T \)
11 \( 1 + (3.31 - 0.189i)T \)
good2 \( 1 + (-1.18 - 1.63i)T + (-0.618 + 1.90i)T^{2} \)
3 \( 1 + (2.49 - 0.809i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + (-0.918 - 0.298i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (2.65 + 3.66i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.96 - 2.69i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.01 + 3.11i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 3.36iT - 23T^{2} \)
29 \( 1 + (1.51 - 4.67i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.338 + 0.245i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-6.02 - 1.95i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.78 - 5.50i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 2.26iT - 43T^{2} \)
47 \( 1 + (4.11 - 1.33i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.56 + 2.15i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.12 + 9.62i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.99 - 1.45i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 9.60iT - 67T^{2} \)
71 \( 1 + (-4.41 - 3.20i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.36 + 0.443i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-0.812 + 0.590i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (4.34 - 5.98i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 + (-1.77 - 2.44i)T + (-29.9 + 92.2i)T^{2} \)
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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

−15.70546268782328190875260351283, −14.80287771655544750762719209779, −13.21683344733930552302756008189, −12.64186054592978258321359749191, −11.04007970902835482492815432522, −9.941182213386016534626873590292, −7.993057430355807438119123516703, −6.36823692930961163414796606612, −5.30017665309608384250196243583, −4.81495265516526242350699556021, 2.26510552391083776004693714983, 4.67323684807931577230985221630, 5.87366736742610421026383278083, 7.28174496388398372362745796788, 9.979333175689821502076189114924, 10.90680175539321177175561161706, 11.60908900541526552881406373599, 12.61102114368478549316821719926, 13.58146254866192038229845449995, 14.59293524811718969412070412879

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