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Properties

Label 2-495-165.164-c1-0-12
Degree $2$
Conductor $495$
Sign $0.670 - 0.742i$
Analytic cond. $3.95259$
Root an. cond. $1.98811$
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  + i·2-s + 4-s + (2.12 − 0.707i)5-s + 1.41·7-s + 3i·8-s + (0.707 + 2.12i)10-s + (−3 − 1.41i)11-s + 2.82·13-s + 1.41i·14-s − 16-s + 2i·17-s − 4.24i·19-s + (2.12 − 0.707i)20-s + (1.41 − 3i)22-s + (3.99 − 3i)25-s + 2.82i·26-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.5·4-s + (0.948 − 0.316i)5-s + 0.534·7-s + 1.06i·8-s + (0.223 + 0.670i)10-s + (−0.904 − 0.426i)11-s + 0.784·13-s + 0.377i·14-s − 0.250·16-s + 0.485i·17-s − 0.973i·19-s + (0.474 − 0.158i)20-s + (0.301 − 0.639i)22-s + (0.799 − 0.600i)25-s + 0.554i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.670 - 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.670 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.670 - 0.742i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (494, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ 0.670 - 0.742i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82980 + 0.813365i\)
\(L(\frac12)\) \(\approx\) \(1.82980 + 0.813365i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-2.12 + 0.707i)T \)
11 \( 1 + (3 + 1.41i)T \)
good2 \( 1 - iT - 2T^{2} \)
7 \( 1 - 1.41T + 7T^{2} \)
13 \( 1 - 2.82T + 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 4.24iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 1.41T + 43T^{2} \)
47 \( 1 + 8.48T + 47T^{2} \)
53 \( 1 - 4.24T + 53T^{2} \)
59 \( 1 - 8.48iT - 59T^{2} \)
61 \( 1 - 8.48iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 8.48iT - 71T^{2} \)
73 \( 1 + 11.3T + 73T^{2} \)
79 \( 1 + 12.7iT - 79T^{2} \)
83 \( 1 + 14iT - 83T^{2} \)
89 \( 1 + 7.07iT - 89T^{2} \)
97 \( 1 - 12iT - 97T^{2} \)
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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

−10.98095484653471131773501853320, −10.30902600910625413624050038196, −9.023149082366503078022690941621, −8.312511821487596330059000767855, −7.41800517841496954804524772264, −6.29898108670478545266487672215, −5.65685957113597775995537996193, −4.75343028587308086060390330084, −2.90484093324287141952658479576, −1.67467689156553193225982738531, 1.57446289069492053628783392948, 2.49791688873348899494656363465, 3.71169139225288300872463514197, 5.22875316155262699083564508933, 6.13051264240315417658785441786, 7.15959768292360751027679259297, 8.092037282426397733116327584277, 9.425880497560166975624250107104, 10.08451274780536077475789158748, 10.91686531564124369779153275367

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