Properties

Label 2-275-55.32-c1-0-8
Degree $2$
Conductor $275$
Sign $0.757 - 0.652i$
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  + (−1.58 + 1.58i)2-s + (1 − i)3-s − 3.00i·4-s + 3.16i·6-s + (1.58 + 1.58i)8-s + i·9-s + (1 − 3.16i)11-s + (−3.00 − 3.00i)12-s + (3.16 + 3.16i)13-s + 0.999·16-s + (3.16 − 3.16i)17-s + (−1.58 − 1.58i)18-s + 6.32·19-s + (3.41 + 6.58i)22-s + (1 − i)23-s + 3.16·24-s + ⋯
L(s)  = 1  + (−1.11 + 1.11i)2-s + (0.577 − 0.577i)3-s − 1.50i·4-s + 1.29i·6-s + (0.559 + 0.559i)8-s + 0.333i·9-s + (0.301 − 0.953i)11-s + (−0.866 − 0.866i)12-s + (0.877 + 0.877i)13-s + 0.249·16-s + (0.766 − 0.766i)17-s + (−0.372 − 0.372i)18-s + 1.45·19-s + (0.728 + 1.40i)22-s + (0.208 − 0.208i)23-s + 0.645·24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 - 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.757 - 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.757 - 0.652i$
Analytic conductor: \(2.19588\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1/2),\ 0.757 - 0.652i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.863704 + 0.320643i\)
\(L(\frac12)\) \(\approx\) \(0.863704 + 0.320643i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 3.16i)T \)
good2 \( 1 + (1.58 - 1.58i)T - 2iT^{2} \)
3 \( 1 + (-1 + i)T - 3iT^{2} \)
7 \( 1 - 7iT^{2} \)
13 \( 1 + (-3.16 - 3.16i)T + 13iT^{2} \)
17 \( 1 + (-3.16 + 3.16i)T - 17iT^{2} \)
19 \( 1 - 6.32T + 19T^{2} \)
23 \( 1 + (-1 + i)T - 23iT^{2} \)
29 \( 1 + 6.32T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + (3 + 3i)T + 37iT^{2} \)
41 \( 1 - 6.32iT - 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + (3 + 3i)T + 47iT^{2} \)
53 \( 1 + (-1 + i)T - 53iT^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 + 6.32iT - 61T^{2} \)
67 \( 1 + (3 + 3i)T + 67iT^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (3.16 + 3.16i)T + 73iT^{2} \)
79 \( 1 + 6.32T + 79T^{2} \)
83 \( 1 + (-6.32 - 6.32i)T + 83iT^{2} \)
89 \( 1 + 6iT - 89T^{2} \)
97 \( 1 + (-7 - 7i)T + 97iT^{2} \)
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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

−11.84171154091767834979689652131, −10.87716685889400041578604727228, −9.570288476691893119071848549966, −8.937256298501993275796680123856, −8.036013422726171460484632312321, −7.37581355270162389275710725808, −6.41459832462566557158604211094, −5.34008849130754469357084379456, −3.29893448568374250541462654086, −1.28410445897029585626401370469, 1.36153252467255927321820151422, 3.04326589036324223350605420569, 3.84685670890152207854793355263, 5.64613102019009415123944578493, 7.36827518574286363005329649422, 8.362567358667186050037969447665, 9.177183895471643796048389612932, 9.922438502340660117658675414350, 10.49070084420388065971270976647, 11.64804699947195207121789963304

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