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Properties

Label 2-275-275.31-c1-0-3
Degree $2$
Conductor $275$
Sign $0.634 - 0.773i$
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  − 1.47·2-s + (−0.500 + 1.53i)3-s + 0.164·4-s + (−0.844 − 2.07i)5-s + (0.736 − 2.26i)6-s + (−1.36 − 0.991i)7-s + 2.70·8-s + (0.306 + 0.222i)9-s + (1.24 + 3.04i)10-s + (3.31 − 0.00241i)11-s + (−0.0822 + 0.253i)12-s + (0.766 + 0.557i)13-s + (2.00 + 1.45i)14-s + (3.61 − 0.264i)15-s − 4.30·16-s + (−2.10 + 6.48i)17-s + ⋯
L(s)  = 1  − 1.04·2-s + (−0.288 + 0.889i)3-s + 0.0822·4-s + (−0.377 − 0.925i)5-s + (0.300 − 0.924i)6-s + (−0.515 − 0.374i)7-s + 0.954·8-s + (0.102 + 0.0741i)9-s + (0.392 + 0.963i)10-s + (0.999 − 0.000729i)11-s + (−0.0237 + 0.0730i)12-s + (0.212 + 0.154i)13-s + (0.536 + 0.389i)14-s + (0.932 − 0.0683i)15-s − 1.07·16-s + (−0.511 + 1.57i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 - 0.773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.634 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.634 - 0.773i$
Analytic conductor: \(2.19588\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1/2),\ 0.634 - 0.773i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.530361 + 0.250979i\)
\(L(\frac12)\) \(\approx\) \(0.530361 + 0.250979i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.844 + 2.07i)T \)
11 \( 1 + (-3.31 + 0.00241i)T \)
good2 \( 1 + 1.47T + 2T^{2} \)
3 \( 1 + (0.500 - 1.53i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (1.36 + 0.991i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.766 - 0.557i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.10 - 6.48i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 - 1.90T + 19T^{2} \)
23 \( 1 + (-1.35 - 4.16i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 - 5.85T + 29T^{2} \)
31 \( 1 + (-7.43 + 5.40i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-6.42 + 4.67i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-3.14 - 9.66i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 8.15T + 43T^{2} \)
47 \( 1 + (-1.63 + 5.02i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.71 - 5.27i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-1.22 + 0.888i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (9.36 - 6.80i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (3.44 + 2.49i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (0.393 + 0.285i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.38 - 13.5i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (2.91 + 8.97i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.05 + 6.32i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-2.31 - 7.12i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (4.84 + 14.9i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.70863439998556997156284818241, −10.80830662776052306039282994736, −9.884657375305300386909842096060, −9.314948920329145917858228478462, −8.456666165296819278525220848369, −7.47303602765833127344240224430, −6.04038317359907279885332659579, −4.47924885867612324085125253675, −4.01850838954551888614110999698, −1.21135689941596793366133127183, 0.849893895142555609507232306388, 2.73736408388505639466742552014, 4.43295757186533062260677983454, 6.35195903291495253335345520155, 6.92629505187741363050352661668, 7.78785628594247319002705741011, 8.957908740217392284136403662611, 9.726087902362635155323292893075, 10.74737542593060811587495834091, 11.72214241180922606618612446943

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