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Properties

Label 2-65-65.33-c1-0-4
Degree $2$
Conductor $65$
Sign $0.544 + 0.838i$
Analytic cond. $0.519027$
Root an. cond. $0.720435$
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  + (0.759 − 1.31i)2-s + (−0.653 + 0.175i)3-s + (−0.152 − 0.263i)4-s + (0.600 − 2.15i)5-s + (−0.265 + 0.991i)6-s + (−2.24 + 1.29i)7-s + 2.57·8-s + (−2.20 + 1.27i)9-s + (−2.37 − 2.42i)10-s + (1.29 + 4.82i)11-s + (0.145 + 0.145i)12-s + (−2.71 − 2.37i)13-s + 3.93i·14-s + (−0.0150 + 1.51i)15-s + (2.25 − 3.91i)16-s + (0.0211 − 0.0790i)17-s + ⋯
L(s)  = 1  + (0.536 − 0.929i)2-s + (−0.377 + 0.101i)3-s + (−0.0761 − 0.131i)4-s + (0.268 − 0.963i)5-s + (−0.108 + 0.404i)6-s + (−0.849 + 0.490i)7-s + 0.910·8-s + (−0.733 + 0.423i)9-s + (−0.751 − 0.766i)10-s + (0.390 + 1.45i)11-s + (0.0420 + 0.0420i)12-s + (−0.752 − 0.658i)13-s + 1.05i·14-s + (−0.00389 + 0.390i)15-s + (0.564 − 0.977i)16-s + (0.00513 − 0.0191i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.544 + 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.544 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.544 + 0.838i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ 0.544 + 0.838i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.917186 - 0.498263i\)
\(L(\frac12)\) \(\approx\) \(0.917186 - 0.498263i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.600 + 2.15i)T \)
13 \( 1 + (2.71 + 2.37i)T \)
good2 \( 1 + (-0.759 + 1.31i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.653 - 0.175i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (2.24 - 1.29i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.29 - 4.82i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.0211 + 0.0790i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.71 - 0.726i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.05 + 3.91i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (4.31 + 2.49i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.32 + 2.32i)T + 31iT^{2} \)
37 \( 1 + (-0.494 - 0.285i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-10.0 + 2.69i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-0.132 - 0.0354i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 2.30iT - 47T^{2} \)
53 \( 1 + (-6.70 - 6.70i)T + 53iT^{2} \)
59 \( 1 + (-0.694 + 2.59i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.74 + 4.74i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.89 - 13.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.98 - 7.42i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + 6.61T + 73T^{2} \)
79 \( 1 + 5.71iT - 79T^{2} \)
83 \( 1 - 3.70iT - 83T^{2} \)
89 \( 1 + (-17.2 + 4.63i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (2.68 + 4.65i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.49676084193122492999134728214, −13.11485033282463408454289073968, −12.45484994485950886283894627703, −11.77463061186007829713259303022, −10.28930772383150041884934406822, −9.316950714347561335581648555360, −7.61239207557268021759509997968, −5.66348569059169321692111375105, −4.43174507198730003233720311463, −2.45563558054144216882036351767, 3.46011406496183997166334544053, 5.64180863168284094285634145114, 6.44296858380033372851696212504, 7.38584523473876882591642653271, 9.335812257584443146579616121686, 10.70776729057727780767435295308, 11.64475598949573218530670366014, 13.38942326772733521367808873440, 14.13850035343717536952491172512, 14.87698177876564829709138116847

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