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Properties

Label 2-495-55.49-c1-0-4
Degree $2$
Conductor $495$
Sign $-0.123 - 0.992i$
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  + (0.782 − 0.254i)2-s + (−1.06 + 0.777i)4-s + (1.29 + 1.82i)5-s + (1.01 + 1.39i)7-s + (−1.60 + 2.21i)8-s + (1.47 + 1.09i)10-s + (−3.28 − 0.449i)11-s + (−5.67 + 1.84i)13-s + (1.14 + 0.833i)14-s + (0.121 − 0.374i)16-s + (−0.0133 − 0.00432i)17-s + (4.16 + 3.02i)19-s + (−2.80 − 0.938i)20-s + (−2.68 + 0.483i)22-s − 2.73i·23-s + ⋯
L(s)  = 1  + (0.553 − 0.179i)2-s + (−0.534 + 0.388i)4-s + (0.580 + 0.814i)5-s + (0.382 + 0.526i)7-s + (−0.568 + 0.782i)8-s + (0.467 + 0.346i)10-s + (−0.990 − 0.135i)11-s + (−1.57 + 0.511i)13-s + (0.306 + 0.222i)14-s + (0.0303 − 0.0935i)16-s + (−0.00322 − 0.00104i)17-s + (0.956 + 0.694i)19-s + (−0.627 − 0.209i)20-s + (−0.572 + 0.103i)22-s − 0.570i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.123 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.123 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.123 - 0.992i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ -0.123 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.967241 + 1.09477i\)
\(L(\frac12)\) \(\approx\) \(0.967241 + 1.09477i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.29 - 1.82i)T \)
11 \( 1 + (3.28 + 0.449i)T \)
good2 \( 1 + (-0.782 + 0.254i)T + (1.61 - 1.17i)T^{2} \)
7 \( 1 + (-1.01 - 1.39i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (5.67 - 1.84i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.0133 + 0.00432i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-4.16 - 3.02i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 2.73iT - 23T^{2} \)
29 \( 1 + (-7.33 + 5.32i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.32 - 7.16i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-4.02 - 5.53i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-0.686 - 0.498i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 5.87iT - 43T^{2} \)
47 \( 1 + (4.68 - 6.44i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-5.50 + 1.78i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.40 + 4.65i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.31 + 4.04i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 6.02iT - 67T^{2} \)
71 \( 1 + (-2.42 + 7.46i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-6.74 - 9.28i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.512 + 1.57i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (15.5 + 5.05i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 - 8.00T + 89T^{2} \)
97 \( 1 + (-11.4 + 3.72i)T + (78.4 - 57.0i)T^{2} \)
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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

−11.42396556746526743114226553323, −10.15194531332512146038244448109, −9.648553309847449177857022002513, −8.396677603995415548912992445949, −7.65341267073522325883842410867, −6.44326685812828551986135287588, −5.29902731506782100714862728475, −4.67666652293820685479376844863, −3.07500008247054751585221001716, −2.38383120335865837625252894844, 0.74257634186944891066333009846, 2.60573321105109222484917898051, 4.27698620887198652513429350419, 5.14067101057450045764842077323, 5.52786661830867775547587506990, 7.02347135399619868937157928798, 7.942091700256584155279835678535, 9.066611657708999054604400808889, 9.886395623168674546847448967702, 10.40250102989573282740053059237

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