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Properties

Label 2-275-11.4-c1-0-0
Degree $2$
Conductor $275$
Sign $-0.698 - 0.716i$
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  + (−1.06 − 0.776i)2-s + (−0.626 + 1.92i)3-s + (−0.0782 − 0.240i)4-s + (2.16 − 1.57i)6-s + (0.107 + 0.331i)7-s + (−0.920 + 2.83i)8-s + (−0.895 − 0.650i)9-s + (−3.25 − 0.638i)11-s + 0.513·12-s + (−1.50 − 1.09i)13-s + (0.142 − 0.437i)14-s + (2.77 − 2.01i)16-s + (−1.45 + 1.05i)17-s + (0.451 + 1.39i)18-s + (−2.21 + 6.80i)19-s + ⋯
L(s)  = 1  + (−0.756 − 0.549i)2-s + (−0.361 + 1.11i)3-s + (−0.0391 − 0.120i)4-s + (0.884 − 0.642i)6-s + (0.0406 + 0.125i)7-s + (−0.325 + 1.00i)8-s + (−0.298 − 0.216i)9-s + (−0.981 − 0.192i)11-s + 0.148·12-s + (−0.416 − 0.302i)13-s + (0.0380 − 0.116i)14-s + (0.693 − 0.503i)16-s + (−0.353 + 0.256i)17-s + (0.106 + 0.327i)18-s + (−0.507 + 1.56i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.698 - 0.716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.698 - 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.698 - 0.716i$
Analytic conductor: \(2.19588\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1/2),\ -0.698 - 0.716i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.117287 + 0.278111i\)
\(L(\frac12)\) \(\approx\) \(0.117287 + 0.278111i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.25 + 0.638i)T \)
good2 \( 1 + (1.06 + 0.776i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.626 - 1.92i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (-0.107 - 0.331i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (1.50 + 1.09i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.45 - 1.05i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (2.21 - 6.80i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 8.41T + 23T^{2} \)
29 \( 1 + (-1.79 - 5.51i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (4.49 + 3.26i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.571 + 1.75i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.79 - 8.59i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 6.76T + 43T^{2} \)
47 \( 1 + (-2.24 + 6.90i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.277 - 0.201i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.0615 - 0.189i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-6.30 + 4.58i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 12.2T + 67T^{2} \)
71 \( 1 + (7.95 - 5.77i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.313 + 0.963i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (5.36 + 3.89i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (9.00 - 6.53i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 8.84T + 89T^{2} \)
97 \( 1 + (-5.01 - 3.64i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.87661162173971859672409724327, −10.87836711270349800999284512734, −10.25379401027671466285382498455, −9.834240838998695860639961504125, −8.648391426503163125866070612448, −7.80008415080678001326180825289, −5.90119120794669039878740005966, −5.16606424565784946914477086498, −3.86493363338911747409121478746, −2.13996369589842996224693592415, 0.28989486331523643325264834455, 2.35667817053946164131404309565, 4.30697545800362237762158005125, 5.90553052540550947241615488721, 6.97293504272213849111560553732, 7.47188194690386615770381389005, 8.404179393587793071135989759365, 9.437154737140845881710590647248, 10.47921257938063045825114902788, 11.72012289368657350954490159743

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