Properties

Label 2-275-5.4-c1-0-15
Degree $2$
Conductor $275$
Sign $-0.447 - 0.894i$
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  − 2.41i·2-s − 2.82i·3-s − 3.82·4-s − 6.82·6-s + 2i·7-s + 4.41i·8-s − 5.00·9-s + 11-s + 10.8i·12-s − 1.17i·13-s + 4.82·14-s + 2.99·16-s − 6.82i·17-s + 12.0i·18-s + 5.65·21-s − 2.41i·22-s + ⋯
L(s)  = 1  − 1.70i·2-s − 1.63i·3-s − 1.91·4-s − 2.78·6-s + 0.755i·7-s + 1.56i·8-s − 1.66·9-s + 0.301·11-s + 3.12i·12-s − 0.324i·13-s + 1.29·14-s + 0.749·16-s − 1.65i·17-s + 2.84i·18-s + 1.23·21-s − 0.514i·22-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(2.19588\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.537451 + 0.869615i\)
\(L(\frac12)\) \(\approx\) \(0.537451 + 0.869615i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good2 \( 1 + 2.41iT - 2T^{2} \)
3 \( 1 + 2.82iT - 3T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
13 \( 1 + 1.17iT - 13T^{2} \)
17 \( 1 + 6.82iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 2.82iT - 23T^{2} \)
29 \( 1 - 3.65T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 7.65iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 6iT - 43T^{2} \)
47 \( 1 + 2.82iT - 47T^{2} \)
53 \( 1 - 11.6iT - 53T^{2} \)
59 \( 1 + 1.65T + 59T^{2} \)
61 \( 1 + 9.31T + 61T^{2} \)
67 \( 1 + 12.4iT - 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 + 1.17iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + 3.65iT - 97T^{2} \)
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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

−11.71748719119311424796021987103, −10.62223757310466014569792610859, −9.382926957633989313241199957398, −8.609024649444682574509224626087, −7.44566097885244933517184743280, −6.25088904946254021683436791533, −4.86351319736955812035021954396, −3.02196510664762819355575242138, −2.21394959974488844257857504394, −0.824723059246608476425168087687, 3.80125312101073682934274451959, 4.42235240349404129175085364339, 5.53736346430189683680820110309, 6.46973436054866681085470473148, 7.70147510800215435011818007093, 8.663498480924810325294990729435, 9.464759372223001050603872617831, 10.34637056010627434359659982229, 11.23081722949860818297754489195, 12.85825471891489269932012584455

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