Properties

Label 2-275-5.4-c5-0-19
Degree $2$
Conductor $275$
Sign $0.447 + 0.894i$
Analytic cond. $44.1055$
Root an. cond. $6.64120$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.96i·2-s + 22.6i·3-s − 16.4·4-s − 157.·6-s + 169. i·7-s + 108. i·8-s − 270.·9-s − 121·11-s − 373. i·12-s + 25.3i·13-s − 1.17e3·14-s − 1.27e3·16-s + 2.01e3i·17-s − 1.88e3i·18-s + 773.·19-s + ⋯
L(s)  = 1  + 1.23i·2-s + 1.45i·3-s − 0.515·4-s − 1.79·6-s + 1.30i·7-s + 0.596i·8-s − 1.11·9-s − 0.301·11-s − 0.748i·12-s + 0.0415i·13-s − 1.60·14-s − 1.24·16-s + 1.69i·17-s − 1.37i·18-s + 0.491·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(44.1055\)
Root analytic conductor: \(6.64120\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :5/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.760885740\)
\(L(\frac12)\) \(\approx\) \(1.760885740\)
\(L(\frac{7}{2})\) 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 \( 1 + 121T \)
good2 \( 1 - 6.96iT - 32T^{2} \)
3 \( 1 - 22.6iT - 243T^{2} \)
7 \( 1 - 169. iT - 1.68e4T^{2} \)
13 \( 1 - 25.3iT - 3.71e5T^{2} \)
17 \( 1 - 2.01e3iT - 1.41e6T^{2} \)
19 \( 1 - 773.T + 2.47e6T^{2} \)
23 \( 1 + 541. iT - 6.43e6T^{2} \)
29 \( 1 - 5.88e3T + 2.05e7T^{2} \)
31 \( 1 + 915.T + 2.86e7T^{2} \)
37 \( 1 + 1.13e4iT - 6.93e7T^{2} \)
41 \( 1 + 1.54e4T + 1.15e8T^{2} \)
43 \( 1 - 6.09e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.51e4iT - 2.29e8T^{2} \)
53 \( 1 - 1.04e4iT - 4.18e8T^{2} \)
59 \( 1 - 5.02e4T + 7.14e8T^{2} \)
61 \( 1 - 4.55e4T + 8.44e8T^{2} \)
67 \( 1 + 1.12e4iT - 1.35e9T^{2} \)
71 \( 1 - 6.47e4T + 1.80e9T^{2} \)
73 \( 1 - 7.77e4iT - 2.07e9T^{2} \)
79 \( 1 - 8.78e4T + 3.07e9T^{2} \)
83 \( 1 - 1.84e4iT - 3.93e9T^{2} \)
89 \( 1 + 5.26e4T + 5.58e9T^{2} \)
97 \( 1 - 3.87e4iT - 8.58e9T^{2} \)
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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

−11.66143977432764626404351767656, −10.69133243892876813547042644195, −9.768237259969709499147904816121, −8.687264319036477889758815866967, −8.268560541657674500556565324955, −6.71313551484490501521785407999, −5.64460832651666399773488690806, −5.11147984227265158526791832193, −3.81843355478562444177758666712, −2.34749862260797622877571766056, 0.53589138495311689193738463589, 1.12533471612724064963044426396, 2.37452896266112202917931177522, 3.42045734274824107505052491634, 4.88435959819585329934863285187, 6.70544542522439473464537950468, 7.14303176296871730203644465042, 8.175259310710966379661176197449, 9.662093543428621866257395769603, 10.41733823802481703908649593979

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