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Properties

Label 2-275-11.4-c1-0-4
Degree $2$
Conductor $275$
Sign $-0.788 - 0.614i$
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  + (1.63 + 1.18i)2-s + (−0.809 + 2.49i)3-s + (0.647 + 1.99i)4-s + (−4.29 + 3.11i)6-s + (0.298 + 0.918i)7-s + (−0.0595 + 0.183i)8-s + (−3.12 − 2.27i)9-s + (−3.31 − 0.189i)11-s − 5.49·12-s + (3.66 + 2.65i)13-s + (−0.603 + 1.85i)14-s + (3.07 − 2.23i)16-s + (2.69 − 1.96i)17-s + (−2.41 − 7.44i)18-s + (1.01 − 3.11i)19-s + ⋯
L(s)  = 1  + (1.15 + 0.841i)2-s + (−0.467 + 1.43i)3-s + (0.323 + 0.996i)4-s + (−1.75 + 1.27i)6-s + (0.112 + 0.347i)7-s + (−0.0210 + 0.0648i)8-s + (−1.04 − 0.757i)9-s + (−0.998 − 0.0572i)11-s − 1.58·12-s + (1.01 + 0.737i)13-s + (−0.161 + 0.496i)14-s + (0.768 − 0.558i)16-s + (0.654 − 0.475i)17-s + (−0.570 − 1.75i)18-s + (0.232 − 0.715i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 - 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.788 - 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.788 - 0.614i$
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: \((2,\ 275,\ (\ :1/2),\ -0.788 - 0.614i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.632830 + 1.84204i\)
\(L(\frac12)\) \(\approx\) \(0.632830 + 1.84204i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.31 + 0.189i)T \)
good2 \( 1 + (-1.63 - 1.18i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.809 - 2.49i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (-0.298 - 0.918i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-3.66 - 2.65i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.69 + 1.96i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.01 + 3.11i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 3.36T + 23T^{2} \)
29 \( 1 + (-1.51 - 4.67i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.338 - 0.245i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.95 - 6.02i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.78 + 5.50i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 2.26T + 43T^{2} \)
47 \( 1 + (-1.33 + 4.11i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-2.15 - 1.56i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.12 + 9.62i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.99 + 1.45i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 9.60T + 67T^{2} \)
71 \( 1 + (-4.41 + 3.20i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.443 - 1.36i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (0.812 + 0.590i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (5.98 - 4.34i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 12.1T + 89T^{2} \)
97 \( 1 + (-2.44 - 1.77i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.30875361669846668096535340780, −11.36233464780402505674761645503, −10.43527801724498256294824911229, −9.542094405218379472810446164616, −8.371999453372257984990945866770, −7.00037311118589519258632123271, −5.80518909109479622328173659078, −5.16801216453007885119829796792, −4.30332517529468013650245817438, −3.23316617100259924738788111020, 1.30367025921154396788616027944, 2.66871231753699609650735386782, 4.03033010007937574155635535886, 5.57560307468741288731835649773, 6.07586336647953913270150676694, 7.64951933533000909828904004373, 8.154016298530555672183671247874, 10.21475241176205291307325549511, 10.93961846685067281413221786610, 11.87408133961494299983365070371

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