Properties

Label 2-65-65.47-c1-0-4
Degree $2$
Conductor $65$
Sign $-0.506 + 0.862i$
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  − 2.31i·2-s + (−0.240 + 0.240i)3-s − 3.36·4-s + (−1.55 − 1.60i)5-s + (0.556 + 0.556i)6-s + 3.95·7-s + 3.16i·8-s + 2.88i·9-s + (−3.71 + 3.60i)10-s + (−0.556 + 0.556i)11-s + (0.808 − 0.808i)12-s + (3.60 − 0.0370i)13-s − 9.16i·14-s + (0.759 + 0.0117i)15-s + 0.593·16-s + (−1.16 + 1.16i)17-s + ⋯
L(s)  = 1  − 1.63i·2-s + (−0.138 + 0.138i)3-s − 1.68·4-s + (−0.696 − 0.717i)5-s + (0.227 + 0.227i)6-s + 1.49·7-s + 1.11i·8-s + 0.961i·9-s + (−1.17 + 1.14i)10-s + (−0.167 + 0.167i)11-s + (0.233 − 0.233i)12-s + (0.999 − 0.0102i)13-s − 2.45i·14-s + (0.196 + 0.00302i)15-s + 0.148·16-s + (−0.281 + 0.281i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.506 + 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.506 + 0.862i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ -0.506 + 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.409966 - 0.716172i\)
\(L(\frac12)\) \(\approx\) \(0.409966 - 0.716172i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.55 + 1.60i)T \)
13 \( 1 + (-3.60 + 0.0370i)T \)
good2 \( 1 + 2.31iT - 2T^{2} \)
3 \( 1 + (0.240 - 0.240i)T - 3iT^{2} \)
7 \( 1 - 3.95T + 7T^{2} \)
11 \( 1 + (0.556 - 0.556i)T - 11iT^{2} \)
17 \( 1 + (1.16 - 1.16i)T - 17iT^{2} \)
19 \( 1 + (1.24 - 1.24i)T - 19iT^{2} \)
23 \( 1 + (2.80 + 2.80i)T + 23iT^{2} \)
29 \( 1 - 3.47iT - 29T^{2} \)
31 \( 1 + (2.07 + 2.07i)T + 31iT^{2} \)
37 \( 1 + 6.84T + 37T^{2} \)
41 \( 1 + (-7.52 - 7.52i)T + 41iT^{2} \)
43 \( 1 + (7.03 + 7.03i)T + 43iT^{2} \)
47 \( 1 - 9.09T + 47T^{2} \)
53 \( 1 + (-0.958 + 0.958i)T - 53iT^{2} \)
59 \( 1 + (7.46 + 7.46i)T + 59iT^{2} \)
61 \( 1 - 3.68T + 61T^{2} \)
67 \( 1 + 3.78iT - 67T^{2} \)
71 \( 1 + (3.67 + 3.67i)T + 71iT^{2} \)
73 \( 1 + 5.57iT - 73T^{2} \)
79 \( 1 - 9.03iT - 79T^{2} \)
83 \( 1 - 6.16T + 83T^{2} \)
89 \( 1 + (3.51 + 3.51i)T + 89iT^{2} \)
97 \( 1 + 9.03iT - 97T^{2} \)
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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

−14.14308015589348323716134657599, −13.09137093631729181169340632898, −12.04584866376694683923137958945, −11.13473786206505746238446702354, −10.55324615164984181196009859273, −8.810609301095171649520889671554, −7.969873415412816779848593017666, −5.05012988023387437334995194351, −4.06033931094218668455165127899, −1.76097132527405707291291299262, 4.15968694170517662130271696138, 5.74518930177726512045151808561, 6.95738544924012154504552569643, 7.945704458845783226592000082666, 8.880028782471270821226646070878, 10.90729871921244677802393883000, 11.86644662422870613611091275384, 13.67884051500116930358112407090, 14.53363850964500978087954841992, 15.29813697712084519472397903628

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