Properties

Label 2-65-13.6-c2-0-7
Degree $2$
Conductor $65$
Sign $-0.524 + 0.851i$
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  + (−0.590 − 2.20i)2-s + (1.05 − 1.83i)3-s + (−1.03 + 0.598i)4-s + (1.58 − 1.58i)5-s + (−4.66 − 1.24i)6-s + (−0.314 + 1.17i)7-s + (−4.51 − 4.51i)8-s + (2.26 + 3.91i)9-s + (−4.41 − 2.54i)10-s + (−13.3 + 3.57i)11-s + 2.53i·12-s + (12.5 − 3.48i)13-s + 2.76·14-s + (−1.22 − 4.57i)15-s + (−9.67 + 16.7i)16-s + (8.20 − 4.73i)17-s + ⋯
L(s)  = 1  + (−0.295 − 1.10i)2-s + (0.352 − 0.610i)3-s + (−0.259 + 0.149i)4-s + (0.316 − 0.316i)5-s + (−0.776 − 0.208i)6-s + (−0.0449 + 0.167i)7-s + (−0.564 − 0.564i)8-s + (0.251 + 0.435i)9-s + (−0.441 − 0.254i)10-s + (−1.21 + 0.324i)11-s + 0.211i·12-s + (0.963 − 0.268i)13-s + 0.197·14-s + (−0.0816 − 0.304i)15-s + (−0.604 + 1.04i)16-s + (0.482 − 0.278i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.524 + 0.851i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.524 + 0.851i$
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: \((2,\ 65,\ (\ :1),\ -0.524 + 0.851i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.601138 - 1.07654i\)
\(L(\frac12)\) \(\approx\) \(0.601138 - 1.07654i\)
\(L(2)\) 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 \( 1 + (-12.5 + 3.48i)T \)
good2 \( 1 + (0.590 + 2.20i)T + (-3.46 + 2i)T^{2} \)
3 \( 1 + (-1.05 + 1.83i)T + (-4.5 - 7.79i)T^{2} \)
7 \( 1 + (0.314 - 1.17i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (13.3 - 3.57i)T + (104. - 60.5i)T^{2} \)
17 \( 1 + (-8.20 + 4.73i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-23.4 - 6.28i)T + (312. + 180.5i)T^{2} \)
23 \( 1 + (-10.0 - 5.80i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (2.30 - 3.99i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (21.8 - 21.8i)T - 961iT^{2} \)
37 \( 1 + (19.1 - 5.14i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (14.6 + 54.8i)T + (-1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (61.4 - 35.4i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (22.8 + 22.8i)T + 2.20e3iT^{2} \)
53 \( 1 + 3.82T + 2.80e3T^{2} \)
59 \( 1 + (17.8 - 66.5i)T + (-3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (-20.0 - 34.7i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (15.0 + 56.1i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (53.0 + 14.2i)T + (4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (-34.0 - 34.0i)T + 5.32e3iT^{2} \)
79 \( 1 + 27.0T + 6.24e3T^{2} \)
83 \( 1 + (20.0 - 20.0i)T - 6.88e3iT^{2} \)
89 \( 1 + (30.1 - 8.08i)T + (6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (172. + 46.1i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−13.76801639339210633922013325712, −13.00753449660725136167905629892, −12.11225045026067038442568511587, −10.79134351977236995206928876093, −9.919286722789345905314279772345, −8.581327824604601121937757558405, −7.25633874518101859450633578870, −5.42528728565599649862726306127, −3.06124302353320677757258034050, −1.53059359951935151756705139898, 3.25016660738582018696892862180, 5.35430575447105068213350549801, 6.65139508743130324684480115045, 7.902570485547349685137174763161, 9.044040230656589987015258224567, 10.19420882504605807471035999956, 11.48460934032780531126424912118, 13.17455628851027832578077593190, 14.29276358646704039422488825379, 15.29083692524522724239446298719

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