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Properties

Label 2-275-11.4-c1-0-9
Degree $2$
Conductor $275$
Sign $0.966 + 0.255i$
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  + (0.998 + 0.725i)2-s + (0.112 − 0.346i)3-s + (−0.147 − 0.453i)4-s + (0.363 − 0.264i)6-s + (−0.798 − 2.45i)7-s + (0.944 − 2.90i)8-s + (2.31 + 1.68i)9-s + (3.12 − 1.12i)11-s − 0.173·12-s + (2.23 + 1.62i)13-s + (0.985 − 3.03i)14-s + (2.27 − 1.65i)16-s + (−3.11 + 2.26i)17-s + (1.09 + 3.36i)18-s + (−0.0857 + 0.264i)19-s + ⋯
L(s)  = 1  + (0.705 + 0.512i)2-s + (0.0649 − 0.199i)3-s + (−0.0737 − 0.226i)4-s + (0.148 − 0.107i)6-s + (−0.301 − 0.929i)7-s + (0.333 − 1.02i)8-s + (0.773 + 0.561i)9-s + (0.940 − 0.339i)11-s − 0.0501·12-s + (0.619 + 0.449i)13-s + (0.263 − 0.810i)14-s + (0.569 − 0.414i)16-s + (−0.755 + 0.549i)17-s + (0.257 + 0.793i)18-s + (−0.0196 + 0.0605i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.255i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.966 + 0.255i$
Analytic conductor: \(2.19588\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1/2),\ 0.966 + 0.255i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83350 - 0.238458i\)
\(L(\frac12)\) \(\approx\) \(1.83350 - 0.238458i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.12 + 1.12i)T \)
good2 \( 1 + (-0.998 - 0.725i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.112 + 0.346i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (0.798 + 2.45i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-2.23 - 1.62i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.11 - 2.26i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.0857 - 0.264i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 8.40T + 23T^{2} \)
29 \( 1 + (-1.02 - 3.16i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.456 + 0.331i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.161 + 0.497i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.57 - 4.86i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 2.54T + 43T^{2} \)
47 \( 1 + (1.52 - 4.68i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-7.05 - 5.12i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.31 - 7.13i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (11.4 - 8.33i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 3.20T + 67T^{2} \)
71 \( 1 + (-6.79 + 4.93i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.02 + 12.3i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (7.85 + 5.70i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.66 - 1.93i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 2.48T + 89T^{2} \)
97 \( 1 + (-8.81 - 6.40i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.07178860895747786266278524650, −10.76413705919457427357633848919, −10.11205564788360576706302770740, −8.987628839032050229268415398455, −7.64152279728711459882958843390, −6.68823582228389188294216155422, −6.02326916224367815198507408506, −4.42742872416721682266996871926, −3.88441716435767249506435041818, −1.45351825844076690202510518963, 2.14902997725055210749328661247, 3.57388536955125010835977295478, 4.37938681863028102427488082150, 5.72483662127444157633965721954, 6.83688012300393925175593408233, 8.245200541027073197856584745754, 9.128241804000805848108522423494, 10.05337690815622198534033182630, 11.35262886041007331558282788418, 12.08314533885069181050020716240

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