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Properties

Label 2-275-55.54-c2-0-3
Degree $2$
Conductor $275$
Sign $-0.515 - 0.856i$
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  − 1.46·2-s − 0.114i·3-s − 1.86·4-s + 0.167i·6-s − 4.56·7-s + 8.56·8-s + 8.98·9-s + (5.89 − 9.28i)11-s + 0.213i·12-s − 16.5·13-s + 6.66·14-s − 5.05·16-s − 17.2·17-s − 13.1·18-s + 35.8i·19-s + ⋯
L(s)  = 1  − 0.730·2-s − 0.0381i·3-s − 0.466·4-s + 0.0278i·6-s − 0.651·7-s + 1.07·8-s + 0.998·9-s + (0.535 − 0.844i)11-s + 0.0178i·12-s − 1.27·13-s + 0.475·14-s − 0.316·16-s − 1.01·17-s − 0.729·18-s + 1.88i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 - 0.856i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.515 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.515 - 0.856i$
Analytic conductor: \(7.49320\)
Root analytic conductor: \(2.73737\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1),\ -0.515 - 0.856i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.201551 + 0.356570i\)
\(L(\frac12)\) \(\approx\) \(0.201551 + 0.356570i\)
\(L(2)\) 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 \( 1 + (-5.89 + 9.28i)T \)
good2 \( 1 + 1.46T + 4T^{2} \)
3 \( 1 + 0.114iT - 9T^{2} \)
7 \( 1 + 4.56T + 49T^{2} \)
13 \( 1 + 16.5T + 169T^{2} \)
17 \( 1 + 17.2T + 289T^{2} \)
19 \( 1 - 35.8iT - 361T^{2} \)
23 \( 1 - 29.3iT - 529T^{2} \)
29 \( 1 + 8.51iT - 841T^{2} \)
31 \( 1 + 26.3T + 961T^{2} \)
37 \( 1 - 44.4iT - 1.36e3T^{2} \)
41 \( 1 - 52.2iT - 1.68e3T^{2} \)
43 \( 1 - 6.77T + 1.84e3T^{2} \)
47 \( 1 - 15.0iT - 2.20e3T^{2} \)
53 \( 1 - 33.1iT - 2.80e3T^{2} \)
59 \( 1 + 51.5T + 3.48e3T^{2} \)
61 \( 1 - 23.1iT - 3.72e3T^{2} \)
67 \( 1 + 113. iT - 4.48e3T^{2} \)
71 \( 1 - 8.00T + 5.04e3T^{2} \)
73 \( 1 - 32.5T + 5.32e3T^{2} \)
79 \( 1 + 52.0iT - 6.24e3T^{2} \)
83 \( 1 + 43.3T + 6.88e3T^{2} \)
89 \( 1 + 73.8T + 7.92e3T^{2} \)
97 \( 1 - 22.0iT - 9.40e3T^{2} \)
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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

−12.00214887568683506009438326495, −10.76147903586623262225554507316, −9.754925526241026911218380376138, −9.429860281476976910341554909066, −8.155287504175759307224126277806, −7.33150753424863853494655196680, −6.14837872757333939321842770897, −4.70370420075297770093879091843, −3.58016260902171640292069919654, −1.52970829332777798699802492519, 0.27559550827180294252060993208, 2.20029074505558710957741269092, 4.19221481041451359231913342403, 4.91369007537528926384074551113, 6.91920437609623446353520919067, 7.22237451549930557721169322908, 8.805548891377915116196477095051, 9.420728742922955200335344450298, 10.12150727889225713268330191043, 11.06872025133458000615788612813

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