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Properties

Label 2-65-13.6-c2-0-5
Degree $2$
Conductor $65$
Sign $0.999 - 0.0279i$
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  + (0.312 + 1.16i)2-s + (1.19 − 2.06i)3-s + (2.20 − 1.27i)4-s + (−1.58 + 1.58i)5-s + (2.77 + 0.744i)6-s + (2.00 − 7.49i)7-s + (5.58 + 5.58i)8-s + (1.65 + 2.87i)9-s + (−2.33 − 1.34i)10-s + (−16.8 + 4.51i)11-s − 6.06i·12-s + (9.78 + 8.56i)13-s + 9.36·14-s + (1.37 + 5.14i)15-s + (0.321 − 0.556i)16-s + (−6.48 + 3.74i)17-s + ⋯
L(s)  = 1  + (0.156 + 0.582i)2-s + (0.397 − 0.688i)3-s + (0.550 − 0.317i)4-s + (−0.316 + 0.316i)5-s + (0.463 + 0.124i)6-s + (0.286 − 1.07i)7-s + (0.698 + 0.698i)8-s + (0.184 + 0.319i)9-s + (−0.233 − 0.134i)10-s + (−1.53 + 0.410i)11-s − 0.505i·12-s + (0.752 + 0.658i)13-s + 0.668·14-s + (0.0919 + 0.343i)15-s + (0.0200 − 0.0347i)16-s + (−0.381 + 0.220i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0279i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0279i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.999 - 0.0279i$
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: \((2,\ 65,\ (\ :1),\ 0.999 - 0.0279i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.56794 + 0.0219103i\)
\(L(\frac12)\) \(\approx\) \(1.56794 + 0.0219103i\)
\(L(2)\) 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 \( 1 + (-9.78 - 8.56i)T \)
good2 \( 1 + (-0.312 - 1.16i)T + (-3.46 + 2i)T^{2} \)
3 \( 1 + (-1.19 + 2.06i)T + (-4.5 - 7.79i)T^{2} \)
7 \( 1 + (-2.00 + 7.49i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (16.8 - 4.51i)T + (104. - 60.5i)T^{2} \)
17 \( 1 + (6.48 - 3.74i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (22.8 + 6.13i)T + (312. + 180.5i)T^{2} \)
23 \( 1 + (20.4 + 11.7i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-9.01 + 15.6i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (39.1 - 39.1i)T - 961iT^{2} \)
37 \( 1 + (-32.5 + 8.72i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (-12.9 - 48.4i)T + (-1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (-39.3 + 22.7i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (2.92 + 2.92i)T + 2.20e3iT^{2} \)
53 \( 1 + 10.5T + 2.80e3T^{2} \)
59 \( 1 + (-21.4 + 79.8i)T + (-3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (13.9 + 24.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (15.6 + 58.3i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (-50.3 - 13.5i)T + (4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (62.5 + 62.5i)T + 5.32e3iT^{2} \)
79 \( 1 + 9.48T + 6.24e3T^{2} \)
83 \( 1 + (3.75 - 3.75i)T - 6.88e3iT^{2} \)
89 \( 1 + (-109. + 29.4i)T + (6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (18.8 + 5.06i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−14.56377264488005196029261645190, −13.72035147246266123849865947791, −12.78339058362633307913236667882, −10.97643888850338397847091755107, −10.48966250524642864940348912607, −8.174011723427028796591298896863, −7.44464069358842268750540105079, −6.45351799709480205017456192775, −4.57344384520775584836310113911, −2.11783698109635692339964021466, 2.57389723541918239725613494617, 4.01265167668656459285336497806, 5.78192901021064927574961293817, 7.81288472460931131113602110854, 8.828798705201109097021059175272, 10.31648965671976332974639256912, 11.21407047510685213912045752670, 12.44017709531871201424904195847, 13.17867492901775365513936613211, 14.98354722487597133680468425973

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