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Properties

Label 2-275-55.14-c1-0-14
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: $\chi_{275} (124, \cdot )$
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

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

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