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Properties

Label 2-275-55.54-c4-0-64
Degree $2$
Conductor $275$
Sign $-0.931 + 0.364i$
Analytic cond. $28.4267$
Root an. cond. $5.33167$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.47·2-s − 3i·3-s + 14·4-s − 16.4i·6-s − 54.7·7-s − 10.9·8-s + 72·9-s + (11 − 120. i)11-s − 42i·12-s − 186.·13-s − 300·14-s − 284·16-s − 230.·17-s + 394.·18-s − 98.5i·19-s + ⋯
L(s)  = 1  + 1.36·2-s − 0.333i·3-s + 0.875·4-s − 0.456i·6-s − 1.11·7-s − 0.171·8-s + 0.888·9-s + (0.0909 − 0.995i)11-s − 0.291i·12-s − 1.10·13-s − 1.53·14-s − 1.10·16-s − 0.795·17-s + 1.21·18-s − 0.273i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 + 0.364i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.931 + 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.931 + 0.364i$
Analytic conductor: \(28.4267\)
Root analytic conductor: \(5.33167\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :2),\ -0.931 + 0.364i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.324229886\)
\(L(\frac12)\) \(\approx\) \(1.324229886\)
\(L(3)\) 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 \( 1 + (-11 + 120. i)T \)
good2 \( 1 - 5.47T + 16T^{2} \)
3 \( 1 + 3iT - 81T^{2} \)
7 \( 1 + 54.7T + 2.40e3T^{2} \)
13 \( 1 + 186.T + 2.85e4T^{2} \)
17 \( 1 + 230.T + 8.35e4T^{2} \)
19 \( 1 + 98.5iT - 1.30e5T^{2} \)
23 \( 1 - 277iT - 2.79e5T^{2} \)
29 \( 1 + 1.27e3iT - 7.07e5T^{2} \)
31 \( 1 + 1.36e3T + 9.23e5T^{2} \)
37 \( 1 + 167iT - 1.87e6T^{2} \)
41 \( 1 - 1.06e3iT - 2.82e6T^{2} \)
43 \( 1 - 1.20e3T + 3.41e6T^{2} \)
47 \( 1 + 1.70e3iT - 4.87e6T^{2} \)
53 \( 1 - 4.52e3iT - 7.89e6T^{2} \)
59 \( 1 - 2.36e3T + 1.21e7T^{2} \)
61 \( 1 + 3.96e3iT - 1.38e7T^{2} \)
67 \( 1 - 2.80e3iT - 2.01e7T^{2} \)
71 \( 1 - 3.39e3T + 2.54e7T^{2} \)
73 \( 1 + 3.31e3T + 2.83e7T^{2} \)
79 \( 1 + 6.09e3iT - 3.89e7T^{2} \)
83 \( 1 - 832.T + 4.74e7T^{2} \)
89 \( 1 - 4.67e3T + 6.27e7T^{2} \)
97 \( 1 + 4.24e3iT - 8.85e7T^{2} \)
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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

−11.18932961471605486940627470584, −9.882205444299047706596340894354, −9.052352114810095724845863587479, −7.46348480824276089771804785491, −6.56719998329854476280908842826, −5.74716638713123096783570202463, −4.49760893072125750686205566207, −3.51058447946788436459781544661, −2.36358324555860989996466946616, −0.23910101244730748511070629134, 2.22168512381166335600852833350, 3.52151218303618039358495849387, 4.43847186159499758629736853713, 5.29284786032966296900098725363, 6.66211944389172203537369667366, 7.19886176353147046283544185682, 9.110889928723443314213087819317, 9.790321096658354997362079947213, 10.78854428997439005400431323286, 12.18303840257515822021785099922

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