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Properties

Label 2-275-11.4-c1-0-1
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: $\chi_{275} (26, \cdot )$
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

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

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