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Properties

Label 2-55-5.4-c1-0-3
Degree $2$
Conductor $55$
Sign $-0.306 + 0.951i$
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  − 2.52i·2-s + 0.792i·3-s − 4.37·4-s + (0.686 − 2.12i)5-s + 2·6-s + 3.46i·7-s + 5.98i·8-s + 2.37·9-s + (−5.37 − 1.73i)10-s − 11-s − 3.46i·12-s + 8.74·14-s + (1.68 + 0.543i)15-s + 6.37·16-s + 1.58i·17-s − 5.98i·18-s + ⋯
L(s)  = 1  − 1.78i·2-s + 0.457i·3-s − 2.18·4-s + (0.306 − 0.951i)5-s + 0.816·6-s + 1.30i·7-s + 2.11i·8-s + 0.790·9-s + (−1.69 − 0.547i)10-s − 0.301·11-s − 1.00i·12-s + 2.33·14-s + (0.435 + 0.140i)15-s + 1.59·16-s + 0.384i·17-s − 1.41i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.306 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.306 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $-0.306 + 0.951i$
Analytic conductor: \(0.439177\)
Root analytic conductor: \(0.662704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :1/2),\ -0.306 + 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.479061 - 0.657795i\)
\(L(\frac12)\) \(\approx\) \(0.479061 - 0.657795i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.686 + 2.12i)T \)
11 \( 1 + T \)
good2 \( 1 + 2.52iT - 2T^{2} \)
3 \( 1 - 0.792iT - 3T^{2} \)
7 \( 1 - 3.46iT - 7T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 1.58iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 0.792iT - 23T^{2} \)
29 \( 1 + 8.74T + 29T^{2} \)
31 \( 1 - 3.37T + 31T^{2} \)
37 \( 1 + 1.08iT - 37T^{2} \)
41 \( 1 - 8.74T + 41T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 + 6.63iT - 47T^{2} \)
53 \( 1 + 10.0iT - 53T^{2} \)
59 \( 1 - 7.37T + 59T^{2} \)
61 \( 1 + 0.744T + 61T^{2} \)
67 \( 1 - 9.30iT - 67T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 + 1.25T + 79T^{2} \)
83 \( 1 + 6.63iT - 83T^{2} \)
89 \( 1 + 1.37T + 89T^{2} \)
97 \( 1 - 5.84iT - 97T^{2} \)
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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

−14.87119647687572433464543991708, −13.13888511353397494631054032513, −12.68066553574276670250761239103, −11.63932991153949792064059311206, −10.34868265170857292905220817206, −9.384390295239582833487040197405, −8.561913929967990611253256812393, −5.42054206796080839394948654022, −4.12586615836761186344194460914, −2.09429720371933036108397787319, 4.27157937693817625799324775125, 6.13306182954689398397886189556, 7.16134975081704456913463737652, 7.72407164836328248452341648958, 9.572150857601680511869988955596, 10.76708718026809165163786206272, 13.02534180515613867434447371242, 13.73305760811598571137660315050, 14.64086599771379449077563887432, 15.62265946182949739567442883376

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