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Properties

Label 2-65-13.6-c2-0-8
Degree $2$
Conductor $65$
Sign $-0.0421 + 0.999i$
Analytic cond. $1.77112$
Root an. cond. $1.33083$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.305 − 1.14i)2-s + (2.30 − 3.98i)3-s + (2.25 − 1.30i)4-s + (−1.58 + 1.58i)5-s + (−5.25 − 1.40i)6-s + (−3.05 + 11.4i)7-s + (−5.51 − 5.51i)8-s + (−6.08 − 10.5i)9-s + (2.28 + 1.32i)10-s + (16.5 − 4.43i)11-s − 11.9i·12-s + (−9.22 + 9.15i)13-s + 13.9·14-s + (2.66 + 9.93i)15-s + (0.590 − 1.02i)16-s + (−2.58 + 1.49i)17-s + ⋯
L(s)  = 1  + (−0.152 − 0.570i)2-s + (0.767 − 1.32i)3-s + (0.563 − 0.325i)4-s + (−0.316 + 0.316i)5-s + (−0.875 − 0.234i)6-s + (−0.436 + 1.62i)7-s + (−0.689 − 0.689i)8-s + (−0.676 − 1.17i)9-s + (0.228 + 0.132i)10-s + (1.50 − 0.403i)11-s − 0.998i·12-s + (−0.709 + 0.704i)13-s + 0.997·14-s + (0.177 + 0.662i)15-s + (0.0369 − 0.0639i)16-s + (−0.151 + 0.0877i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0421 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0421 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.0421 + 0.999i$
Analytic conductor: \(1.77112\)
Root analytic conductor: \(1.33083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1),\ -0.0421 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.00234 - 1.04550i\)
\(L(\frac12)\) \(\approx\) \(1.00234 - 1.04550i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.58 - 1.58i)T \)
13 \( 1 + (9.22 - 9.15i)T \)
good2 \( 1 + (0.305 + 1.14i)T + (-3.46 + 2i)T^{2} \)
3 \( 1 + (-2.30 + 3.98i)T + (-4.5 - 7.79i)T^{2} \)
7 \( 1 + (3.05 - 11.4i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (-16.5 + 4.43i)T + (104. - 60.5i)T^{2} \)
17 \( 1 + (2.58 - 1.49i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (7.87 + 2.10i)T + (312. + 180.5i)T^{2} \)
23 \( 1 + (-12.9 - 7.47i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-6.17 + 10.7i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (22.8 - 22.8i)T - 961iT^{2} \)
37 \( 1 + (-44.9 + 12.0i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (-2.59 - 9.67i)T + (-1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (42.2 - 24.3i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (55.7 + 55.7i)T + 2.20e3iT^{2} \)
53 \( 1 + 19.2T + 2.80e3T^{2} \)
59 \( 1 + (2.26 - 8.43i)T + (-3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (-32.1 - 55.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (1.56 + 5.84i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (55.5 + 14.8i)T + (4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (25.0 + 25.0i)T + 5.32e3iT^{2} \)
79 \( 1 + 6.57T + 6.24e3T^{2} \)
83 \( 1 + (26.1 - 26.1i)T - 6.88e3iT^{2} \)
89 \( 1 + (-17.3 + 4.63i)T + (6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (-121. - 32.6i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−14.52511300175756062658262206028, −12.96530293868076388286730783766, −12.00840266929612033571733284234, −11.51308426864332724351338950300, −9.489157917996490543100479047806, −8.657531438142730567541258994381, −6.97550689533610971205354665775, −6.21715629671314051186593838239, −3.04096601593050246712445727391, −1.85531316974596871934119693953, 3.40101482640906084542656733795, 4.47695825593204629180128919061, 6.73452663080187450368030870634, 7.85603752789314801775495204703, 9.155227186755528370133575725466, 10.15996842263767682177866576191, 11.31505944826737557030239594113, 12.84096970434151301443201120317, 14.41650735008870895983108832110, 14.92600219872301776160426932748

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