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Properties

Label 2-275-11.3-c1-0-9
Degree $2$
Conductor $275$
Sign $0.324 + 0.946i$
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  + (−0.168 + 0.122i)2-s + (−0.585 − 1.80i)3-s + (−0.604 + 1.86i)4-s + (0.319 + 0.231i)6-s + (0.398 − 1.22i)7-s + (−0.254 − 0.783i)8-s + (−0.475 + 0.345i)9-s + (0.898 − 3.19i)11-s + 3.70·12-s + (4.40 − 3.20i)13-s + (0.0829 + 0.255i)14-s + (−3.02 − 2.19i)16-s + (−3.18 − 2.31i)17-s + (0.0378 − 0.116i)18-s + (0.693 + 2.13i)19-s + ⋯
L(s)  = 1  + (−0.119 + 0.0865i)2-s + (−0.337 − 1.04i)3-s + (−0.302 + 0.930i)4-s + (0.130 + 0.0946i)6-s + (0.150 − 0.463i)7-s + (−0.0900 − 0.277i)8-s + (−0.158 + 0.115i)9-s + (0.271 − 0.962i)11-s + 1.06·12-s + (1.22 − 0.888i)13-s + (0.0221 + 0.0682i)14-s + (−0.756 − 0.549i)16-s + (−0.771 − 0.560i)17-s + (0.00892 − 0.0274i)18-s + (0.159 + 0.489i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.324 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.324 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.324 + 0.946i$
Analytic conductor: \(2.19588\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1/2),\ 0.324 + 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.819819 - 0.585730i\)
\(L(\frac12)\) \(\approx\) \(0.819819 - 0.585730i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.898 + 3.19i)T \)
good2 \( 1 + (0.168 - 0.122i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.585 + 1.80i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (-0.398 + 1.22i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-4.40 + 3.20i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (3.18 + 2.31i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.693 - 2.13i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 0.711T + 23T^{2} \)
29 \( 1 + (-1.13 + 3.47i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-5.22 + 3.79i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.55 - 7.85i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.90 - 12.0i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 1.31T + 43T^{2} \)
47 \( 1 + (1.39 + 4.29i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (7.86 - 5.71i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.75 + 8.47i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.48 - 1.08i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 4.30T + 67T^{2} \)
71 \( 1 + (8.21 + 5.97i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.70 - 11.4i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (10.3 - 7.53i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-10.1 - 7.37i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 6.28T + 89T^{2} \)
97 \( 1 + (0.245 - 0.178i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.71692477270146857840547150696, −11.12841431841414873145985958952, −9.714417070911193264880816811016, −8.399341839466841746435362970605, −7.940617135406213439113178853967, −6.80170091433284658409990689806, −5.99447440560002467932310493882, −4.28824219998324538104520404837, −3.05429846024216674251380681545, −0.929260523826921530712593046165, 1.83205248794830684443086280882, 4.04953965490504657232711388367, 4.81675724868211974765100346771, 5.86546961204584027463472957655, 6.95103289334238476626213480848, 8.836083485224735536456590503449, 9.182445435233214540525893574875, 10.33709845733294875981977900234, 10.85197263746534306301657259121, 11.76006403060297775753920342624

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