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Properties

Label 2-65-65.18-c1-0-1
Degree $2$
Conductor $65$
Sign $0.257 - 0.966i$
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  + 1.57i·2-s + (0.725 + 0.725i)3-s − 0.494·4-s + (0.146 − 2.23i)5-s + (−1.14 + 1.14i)6-s − 4.24·7-s + 2.37i·8-s − 1.94i·9-s + (3.52 + 0.231i)10-s + (1.14 + 1.14i)11-s + (−0.358 − 0.358i)12-s + (−0.231 − 3.59i)13-s − 6.71i·14-s + (1.72 − 1.51i)15-s − 4.74·16-s + (4.37 + 4.37i)17-s + ⋯
L(s)  = 1  + 1.11i·2-s + (0.419 + 0.419i)3-s − 0.247·4-s + (0.0654 − 0.997i)5-s + (−0.468 + 0.468i)6-s − 1.60·7-s + 0.840i·8-s − 0.648i·9-s + (1.11 + 0.0731i)10-s + (0.345 + 0.345i)11-s + (−0.103 − 0.103i)12-s + (−0.0641 − 0.997i)13-s − 1.79i·14-s + (0.445 − 0.390i)15-s − 1.18·16-s + (1.06 + 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.257 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.257 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.257 - 0.966i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ 0.257 - 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.771435 + 0.592484i\)
\(L(\frac12)\) \(\approx\) \(0.771435 + 0.592484i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.146 + 2.23i)T \)
13 \( 1 + (0.231 + 3.59i)T \)
good2 \( 1 - 1.57iT - 2T^{2} \)
3 \( 1 + (-0.725 - 0.725i)T + 3iT^{2} \)
7 \( 1 + 4.24T + 7T^{2} \)
11 \( 1 + (-1.14 - 1.14i)T + 11iT^{2} \)
17 \( 1 + (-4.37 - 4.37i)T + 17iT^{2} \)
19 \( 1 + (0.274 + 0.274i)T + 19iT^{2} \)
23 \( 1 + (1.64 - 1.64i)T - 23iT^{2} \)
29 \( 1 - 2.79iT - 29T^{2} \)
31 \( 1 + (2.30 - 2.30i)T - 31iT^{2} \)
37 \( 1 + 2.04T + 37T^{2} \)
41 \( 1 + (0.883 - 0.883i)T - 41iT^{2} \)
43 \( 1 + (-0.944 + 0.944i)T - 43iT^{2} \)
47 \( 1 - 0.483T + 47T^{2} \)
53 \( 1 + (7.24 + 7.24i)T + 53iT^{2} \)
59 \( 1 + (-0.311 + 0.311i)T - 59iT^{2} \)
61 \( 1 - 11.2T + 61T^{2} \)
67 \( 1 - 7.11iT - 67T^{2} \)
71 \( 1 + (-7.42 + 7.42i)T - 71iT^{2} \)
73 \( 1 + 4.96iT - 73T^{2} \)
79 \( 1 - 10.7iT - 79T^{2} \)
83 \( 1 - 11.8T + 83T^{2} \)
89 \( 1 + (1.10 - 1.10i)T - 89iT^{2} \)
97 \( 1 + 10.7iT - 97T^{2} \)
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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

−15.38763577541778002457338258347, −14.41046244316024805183969115421, −12.97756632056295336957325383074, −12.21883796938470752009550606462, −10.12260607566758728403979410742, −9.188259157480587344176076975326, −8.091252702559625621259112100176, −6.58916540207676816303569402893, −5.51807968751059160155402231676, −3.56539913698672882849639035954, 2.47263790372287207439622527512, 3.55001204275113155740720305607, 6.38173171427100255901988622293, 7.33928301644251740460690751262, 9.382587129815723394773152793133, 10.17595471197710671652143581366, 11.33971514368692411554887363185, 12.36486080047663389016895080990, 13.47801584717281754378664346728, 14.22719402427476737197929931733

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