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Properties

Label 2-65-65.38-c2-0-5
Degree $2$
Conductor $65$
Sign $0.222 - 0.975i$
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  + (2.14 + 2.14i)2-s + (1.85 + 1.85i)3-s + 5.18i·4-s + (−1.93 − 4.60i)5-s + 7.96i·6-s + (−5.84 − 5.84i)7-s + (−2.53 + 2.53i)8-s − 2.09i·9-s + (5.72 − 14.0i)10-s + 8.13i·11-s + (−9.62 + 9.62i)12-s + (−0.884 + 12.9i)13-s − 25.0i·14-s + (4.96 − 12.1i)15-s + 9.88·16-s + (−15.0 + 15.0i)17-s + ⋯
L(s)  = 1  + (1.07 + 1.07i)2-s + (0.619 + 0.619i)3-s + 1.29i·4-s + (−0.387 − 0.921i)5-s + 1.32i·6-s + (−0.835 − 0.835i)7-s + (−0.316 + 0.316i)8-s − 0.232i·9-s + (0.572 − 1.40i)10-s + 0.739i·11-s + (−0.802 + 0.802i)12-s + (−0.0680 + 0.997i)13-s − 1.79i·14-s + (0.331 − 0.810i)15-s + 0.617·16-s + (−0.887 + 0.887i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.222 - 0.975i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.222 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.222 - 0.975i$
Analytic conductor: \(1.77112\)
Root analytic conductor: \(1.33083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1),\ 0.222 - 0.975i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.63119 + 1.30148i\)
\(L(\frac12)\) \(\approx\) \(1.63119 + 1.30148i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.93 + 4.60i)T \)
13 \( 1 + (0.884 - 12.9i)T \)
good2 \( 1 + (-2.14 - 2.14i)T + 4iT^{2} \)
3 \( 1 + (-1.85 - 1.85i)T + 9iT^{2} \)
7 \( 1 + (5.84 + 5.84i)T + 49iT^{2} \)
11 \( 1 - 8.13iT - 121T^{2} \)
17 \( 1 + (15.0 - 15.0i)T - 289iT^{2} \)
19 \( 1 - 7.91T + 361T^{2} \)
23 \( 1 + (0.359 + 0.359i)T + 529iT^{2} \)
29 \( 1 + 39.6iT - 841T^{2} \)
31 \( 1 + 10.3iT - 961T^{2} \)
37 \( 1 + (14.3 + 14.3i)T + 1.36e3iT^{2} \)
41 \( 1 - 38.9iT - 1.68e3T^{2} \)
43 \( 1 + (37.8 + 37.8i)T + 1.84e3iT^{2} \)
47 \( 1 + (-56.8 - 56.8i)T + 2.20e3iT^{2} \)
53 \( 1 + (-49.5 - 49.5i)T + 2.80e3iT^{2} \)
59 \( 1 - 37.9T + 3.48e3T^{2} \)
61 \( 1 + 27.8T + 3.72e3T^{2} \)
67 \( 1 + (17.5 + 17.5i)T + 4.48e3iT^{2} \)
71 \( 1 + 106. iT - 5.04e3T^{2} \)
73 \( 1 + (67.9 - 67.9i)T - 5.32e3iT^{2} \)
79 \( 1 + 106. iT - 6.24e3T^{2} \)
83 \( 1 + (-8.61 + 8.61i)T - 6.88e3iT^{2} \)
89 \( 1 + 70.4T + 7.92e3T^{2} \)
97 \( 1 + (-33.7 - 33.7i)T + 9.40e3iT^{2} \)
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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

−15.06340707251889462931437837134, −13.86263609031677045476203868769, −13.10412169702903810545964094791, −12.04758767480591002749036522014, −10.02952896160424181890013058656, −8.940322268657316149244387581459, −7.48830409523991998811723589570, −6.30142129523445920098824869442, −4.46493168616717852358494361732, −3.88910050090593972278272452212, 2.57397942682311281035909280446, 3.30357404022346137261437744925, 5.43455167040525212742046240768, 7.04584966871954773935005470518, 8.558753721978919053525320120017, 10.28344256972433391347084823005, 11.30201800379772649408156552182, 12.34504464599807740863164137728, 13.28962107360958020737109089741, 13.99638327330053155862988839213

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