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Properties

Label 2-275-11.4-c1-0-10
Degree $2$
Conductor $275$
Sign $0.202 + 0.979i$
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  + (0.0756 + 0.0549i)2-s + (−0.453 + 1.39i)3-s + (−0.615 − 1.89i)4-s + (−0.110 + 0.0806i)6-s + (−1.39 − 4.30i)7-s + (0.115 − 0.354i)8-s + (0.686 + 0.498i)9-s + (−2.39 − 2.29i)11-s + 2.92·12-s + (−0.924 − 0.671i)13-s + (0.130 − 0.402i)14-s + (−3.19 + 2.32i)16-s + (2.72 − 1.98i)17-s + (0.0245 + 0.0754i)18-s + (1.88 − 5.78i)19-s + ⋯
L(s)  = 1  + (0.0534 + 0.0388i)2-s + (−0.261 + 0.805i)3-s + (−0.307 − 0.946i)4-s + (−0.0452 + 0.0329i)6-s + (−0.528 − 1.62i)7-s + (0.0407 − 0.125i)8-s + (0.228 + 0.166i)9-s + (−0.723 − 0.690i)11-s + 0.843·12-s + (−0.256 − 0.186i)13-s + (0.0349 − 0.107i)14-s + (−0.798 + 0.580i)16-s + (0.661 − 0.480i)17-s + (0.00578 + 0.0177i)18-s + (0.431 − 1.32i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.202 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.202 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.202 + 0.979i$
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: \((2,\ 275,\ (\ :1/2),\ 0.202 + 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.716880 - 0.583937i\)
\(L(\frac12)\) \(\approx\) \(0.716880 - 0.583937i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.39 + 2.29i)T \)
good2 \( 1 + (-0.0756 - 0.0549i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.453 - 1.39i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (1.39 + 4.30i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (0.924 + 0.671i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.72 + 1.98i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.88 + 5.78i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 5.45T + 23T^{2} \)
29 \( 1 + (-1.02 - 3.15i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.44 + 1.05i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.460 + 1.41i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.539 - 1.66i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 0.263T + 43T^{2} \)
47 \( 1 + (2.13 - 6.58i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.16 + 0.846i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.18 + 6.72i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.02 - 1.47i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 0.516T + 67T^{2} \)
71 \( 1 + (-8.68 + 6.30i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.75 - 5.40i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-9.14 - 6.64i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-3.62 + 2.63i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 + (2.71 + 1.97i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.04628800916005755215629417291, −10.77916486635183925983848003921, −9.901343338118355668582599035783, −9.266875358281918544838120324592, −7.62543017817165213665592676227, −6.71029730432573454161725978822, −5.27322499330269977424667107120, −4.64083677736015246372441103360, −3.31695983544725952882665837666, −0.73803624662342092181972885555, 2.16840713625123104294375783936, 3.43810147092618161030688432072, 5.08270074147069357330463016892, 6.16412048609059406509889259759, 7.30450027976084040747263480837, 8.118963311586024136903348041332, 9.149061242067345826813823612364, 10.04564420835930541621027840203, 11.69185419288344563001557455022, 12.36736989287611428570893347885

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