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Properties

Label 2-275-5.2-c2-0-11
Degree $2$
Conductor $275$
Sign $-0.793 + 0.608i$
Analytic cond. $7.49320$
Root an. cond. $2.73737$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.61 + 2.61i)2-s + (−3.50 + 3.50i)3-s + 9.71i·4-s − 18.3·6-s + (6.89 + 6.89i)7-s + (−14.9 + 14.9i)8-s − 15.5i·9-s + 3.31·11-s + (−34.0 − 34.0i)12-s + (3.74 − 3.74i)13-s + 36.1i·14-s − 39.5·16-s + (8.33 + 8.33i)17-s + (40.7 − 40.7i)18-s − 24.9i·19-s + ⋯
L(s)  = 1  + (1.30 + 1.30i)2-s + (−1.16 + 1.16i)3-s + 2.42i·4-s − 3.05·6-s + (0.985 + 0.985i)7-s + (−1.87 + 1.87i)8-s − 1.72i·9-s + 0.301·11-s + (−2.83 − 2.83i)12-s + (0.287 − 0.287i)13-s + 2.58i·14-s − 2.47·16-s + (0.490 + 0.490i)17-s + (2.26 − 2.26i)18-s − 1.31i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.793 + 0.608i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.793 + 0.608i$
Analytic conductor: \(7.49320\)
Root analytic conductor: \(2.73737\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (232, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1),\ -0.793 + 0.608i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.747041 - 2.20159i\)
\(L(\frac12)\) \(\approx\) \(0.747041 - 2.20159i\)
\(L(2)\) 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 \( 1 - 3.31T \)
good2 \( 1 + (-2.61 - 2.61i)T + 4iT^{2} \)
3 \( 1 + (3.50 - 3.50i)T - 9iT^{2} \)
7 \( 1 + (-6.89 - 6.89i)T + 49iT^{2} \)
13 \( 1 + (-3.74 + 3.74i)T - 169iT^{2} \)
17 \( 1 + (-8.33 - 8.33i)T + 289iT^{2} \)
19 \( 1 + 24.9iT - 361T^{2} \)
23 \( 1 + (-26.1 + 26.1i)T - 529iT^{2} \)
29 \( 1 - 31.2iT - 841T^{2} \)
31 \( 1 + 2.56T + 961T^{2} \)
37 \( 1 + (-7.48 - 7.48i)T + 1.36e3iT^{2} \)
41 \( 1 + 27.3T + 1.68e3T^{2} \)
43 \( 1 + (33.0 - 33.0i)T - 1.84e3iT^{2} \)
47 \( 1 + (17.6 + 17.6i)T + 2.20e3iT^{2} \)
53 \( 1 + (47.0 - 47.0i)T - 2.80e3iT^{2} \)
59 \( 1 + 44.3iT - 3.48e3T^{2} \)
61 \( 1 - 0.515T + 3.72e3T^{2} \)
67 \( 1 + (5.39 + 5.39i)T + 4.48e3iT^{2} \)
71 \( 1 - 75.9T + 5.04e3T^{2} \)
73 \( 1 + (21.1 - 21.1i)T - 5.32e3iT^{2} \)
79 \( 1 + 80.1iT - 6.24e3T^{2} \)
83 \( 1 + (-79.3 + 79.3i)T - 6.88e3iT^{2} \)
89 \( 1 + 104. iT - 7.92e3T^{2} \)
97 \( 1 + (-103. - 103. i)T + 9.40e3iT^{2} \)
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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

−12.25776375248311273144131770030, −11.50712078283006475234144789358, −10.73823333486180490957130907751, −9.115118138416825908429514670598, −8.257240635165975079805979325645, −6.79888489200745681950272925944, −5.94360520092452333453119044200, −4.98279814275232654926542369278, −4.71956601583889069927658811129, −3.25801407472045045259003395983, 0.999601325954366181631593973347, 1.77353090447464919916256880628, 3.68788151610974712101094596694, 4.89272371883311273334866218149, 5.68647985621435402003259833142, 6.76011898856137566939131391687, 7.82848034800124287925357513199, 9.823568735128419485158593977163, 10.82471966706820666178052077756, 11.43176101461983994406469269540

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