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Properties

Label 2-275-55.4-c1-0-1
Degree $2$
Conductor $275$
Sign $-0.999 + 0.00747i$
Analytic cond. $2.19588$
Root an. cond. $1.48185$
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  + (−0.796 + 1.09i)2-s + (0.547 + 0.177i)3-s + (0.0501 + 0.154i)4-s + (−0.631 + 0.458i)6-s + (−3.47 + 1.12i)7-s + (−2.78 − 0.905i)8-s + (−2.15 − 1.56i)9-s + (0.490 + 3.28i)11-s + 0.0933i·12-s + (−1.66 + 2.29i)13-s + (1.52 − 4.70i)14-s + (2.95 − 2.14i)16-s + (2.17 + 2.98i)17-s + (3.44 − 1.11i)18-s + (0.0293 − 0.0904i)19-s + ⋯
L(s)  = 1  + (−0.563 + 0.775i)2-s + (0.315 + 0.102i)3-s + (0.0250 + 0.0771i)4-s + (−0.257 + 0.187i)6-s + (−1.31 + 0.426i)7-s + (−0.985 − 0.320i)8-s + (−0.719 − 0.522i)9-s + (0.147 + 0.989i)11-s + 0.0269i·12-s + (−0.461 + 0.635i)13-s + (0.408 − 1.25i)14-s + (0.738 − 0.536i)16-s + (0.526 + 0.724i)17-s + (0.811 − 0.263i)18-s + (0.00674 − 0.0207i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00747i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.999 + 0.00747i$
Analytic conductor: \(2.19588\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (224, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1/2),\ -0.999 + 0.00747i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00220525 - 0.589733i\)
\(L(\frac12)\) \(\approx\) \(0.00220525 - 0.589733i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 + (-0.490 - 3.28i)T \)
good2 \( 1 + (0.796 - 1.09i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.547 - 0.177i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + (3.47 - 1.12i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.66 - 2.29i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.17 - 2.98i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.0293 + 0.0904i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 1.16iT - 23T^{2} \)
29 \( 1 + (-2.08 - 6.42i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (5.48 + 3.98i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-9.35 + 3.04i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.57 - 7.91i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 2.96iT - 43T^{2} \)
47 \( 1 + (-2.11 - 0.687i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.75 + 2.42i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.62 - 8.09i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-6.86 + 4.98i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 13.4iT - 67T^{2} \)
71 \( 1 + (6.71 - 4.88i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.25 + 0.407i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (11.2 + 8.15i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-6.25 - 8.61i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 + (-2.54 + 3.50i)T + (-29.9 - 92.2i)T^{2} \)
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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

−12.41321062034327102830982866588, −11.61269928707709450967740524061, −9.928968616179725986163965310340, −9.367438909596232677706429556098, −8.629171223179739624400898301589, −7.44577066797481361968739535936, −6.60167059692584976616638858581, −5.74514001317296683980044063716, −3.83918975035262543886545957142, −2.71242486204635727632737602712, 0.48841012474344612065538495176, 2.60968400459934197277580531358, 3.39015195240389787076391756768, 5.44684792965018424306070369826, 6.37246817162784364028072001615, 7.74288048987982410470951713297, 8.811581589301565221043529469843, 9.651096649309886949928050708379, 10.40405986410559247905616987515, 11.25104920699014024535628216758

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