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Properties

Label 2-495-11.9-c1-0-8
Degree $2$
Conductor $495$
Sign $0.996 - 0.0812i$
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  + (0.288 − 0.887i)2-s + (0.912 + 0.663i)4-s + (0.309 + 0.951i)5-s + (1.65 + 1.19i)7-s + (2.36 − 1.71i)8-s + 0.933·10-s + (−1.85 + 2.75i)11-s + (−0.447 + 1.37i)13-s + (1.54 − 1.11i)14-s + (−0.145 − 0.446i)16-s + (−0.267 − 0.824i)17-s + (−2.53 + 1.83i)19-s + (−0.348 + 1.07i)20-s + (1.90 + 2.43i)22-s + 4.70·23-s + ⋯
L(s)  = 1  + (0.203 − 0.627i)2-s + (0.456 + 0.331i)4-s + (0.138 + 0.425i)5-s + (0.624 + 0.453i)7-s + (0.835 − 0.606i)8-s + 0.295·10-s + (−0.558 + 0.829i)11-s + (−0.124 + 0.382i)13-s + (0.411 − 0.299i)14-s + (−0.0362 − 0.111i)16-s + (−0.0649 − 0.200i)17-s + (−0.580 + 0.421i)19-s + (−0.0779 + 0.239i)20-s + (0.406 + 0.519i)22-s + 0.981·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.996 - 0.0812i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ 0.996 - 0.0812i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95979 + 0.0797642i\)
\(L(\frac12)\) \(\approx\) \(1.95979 + 0.0797642i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (1.85 - 2.75i)T \)
good2 \( 1 + (-0.288 + 0.887i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 + (-1.65 - 1.19i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (0.447 - 1.37i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.267 + 0.824i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.53 - 1.83i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 4.70T + 23T^{2} \)
29 \( 1 + (-1.64 - 1.19i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-3.27 + 10.0i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.36 + 2.44i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (0.651 - 0.473i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 2.34T + 43T^{2} \)
47 \( 1 + (-8.39 + 6.10i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.22 - 6.86i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (6.73 + 4.89i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.70 + 8.33i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 3.15T + 67T^{2} \)
71 \( 1 + (3.97 + 12.2i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-11.9 - 8.65i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (5.25 - 16.1i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.89 + 5.84i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 3.77T + 89T^{2} \)
97 \( 1 + (0.598 - 1.84i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.01863616829372498588871174216, −10.33895015459935911961986998970, −9.370731494755768597691673444041, −8.138479706465097638950193981772, −7.34856123908458414432091109971, −6.44327598700602750768277832537, −5.09687450000054124231769060908, −4.06148558656043988505000441499, −2.70172654243589276824799674684, −1.88741093335984853616543246860, 1.27725115677054592444774553769, 2.86483001152841644750447164537, 4.59414973100661421883609689863, 5.29200504435165924433204919870, 6.28726095456468148256341940789, 7.23883740950068722239116778281, 8.133634830686127363806033742099, 8.878116849852222806436245746969, 10.39075846653857154208878472084, 10.74419972180437047571890037827

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