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Properties

Label 2-275-5.4-c1-0-4
Degree $2$
Conductor $275$
Sign $-0.894 - 0.447i$
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  + 1.30i·2-s + 2.30i·3-s + 0.302·4-s − 3·6-s − 0.697i·7-s + 3i·8-s − 2.30·9-s − 11-s + 0.697i·12-s + 5i·13-s + 0.908·14-s − 3.30·16-s − 6.90i·17-s − 3.00i·18-s + 19-s + ⋯
L(s)  = 1  + 0.921i·2-s + 1.32i·3-s + 0.151·4-s − 1.22·6-s − 0.263i·7-s + 1.06i·8-s − 0.767·9-s − 0.301·11-s + 0.201i·12-s + 1.38i·13-s + 0.242·14-s − 0.825·16-s − 1.67i·17-s − 0.707i·18-s + 0.229·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.894 - 0.447i$
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: \((2,\ 275,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.320310 + 1.35685i\)
\(L(\frac12)\) \(\approx\) \(0.320310 + 1.35685i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.30iT - 2T^{2} \)
3 \( 1 - 2.30iT - 3T^{2} \)
7 \( 1 + 0.697iT - 7T^{2} \)
13 \( 1 - 5iT - 13T^{2} \)
17 \( 1 + 6.90iT - 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 + 7.30iT - 23T^{2} \)
29 \( 1 + 0.908T + 29T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 + 2.39iT - 37T^{2} \)
41 \( 1 + 5.60T + 41T^{2} \)
43 \( 1 - 7.21iT - 43T^{2} \)
47 \( 1 - 3iT - 47T^{2} \)
53 \( 1 + 1.30iT - 53T^{2} \)
59 \( 1 - 14.2T + 59T^{2} \)
61 \( 1 + 7.90T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 2.60T + 71T^{2} \)
73 \( 1 + 7.90iT - 73T^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 + 3.51iT - 83T^{2} \)
89 \( 1 + 1.69T + 89T^{2} \)
97 \( 1 + 15.3iT - 97T^{2} \)
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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

−11.97490804885884145918597945887, −11.25971529994801652939818165244, −10.30177341661500974213568475667, −9.399738631690419503626370767692, −8.503618091816844513664367368989, −7.26323469200166635535524155741, −6.41503270745777903545570645784, −5.04780809667250435484143079337, −4.39173213823727211477522426582, −2.70833761404582303362732351325, 1.18188204406964386823878367138, 2.38064580010521595927257501504, 3.59415888423743951655138386265, 5.59839985057147909381453898575, 6.57474469262712668741942296990, 7.62612870580235544207123958487, 8.395217590334114805737362695225, 9.961965054633130232765915284440, 10.63460384639914048019816236784, 11.79420417443743460785915888296

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