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Properties

Label 2-65-65.64-c1-0-0
Degree $2$
Conductor $65$
Sign $-0.124 - 0.992i$
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-s + 2i·3-s − 4-s + (−1 + 2i)5-s − 2i·6-s + 3·8-s − 9-s + (1 − 2i)10-s + 2i·11-s − 2i·12-s + (3 − 2i)13-s + (−4 − 2i)15-s − 16-s + 18-s − 6i·19-s + (1 − 2i)20-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15i·3-s − 0.5·4-s + (−0.447 + 0.894i)5-s − 0.816i·6-s + 1.06·8-s − 0.333·9-s + (0.316 − 0.632i)10-s + 0.603i·11-s − 0.577i·12-s + (0.832 − 0.554i)13-s + (−1.03 − 0.516i)15-s − 0.250·16-s + 0.235·18-s − 1.37i·19-s + (0.223 − 0.447i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.124 - 0.992i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ -0.124 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.365761 + 0.414328i\)
\(L(\frac12)\) \(\approx\) \(0.365761 + 0.414328i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1 - 2i)T \)
13 \( 1 + (-3 + 2i)T \)
good2 \( 1 + T + 2T^{2} \)
3 \( 1 - 2iT - 3T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 6iT - 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + 8iT - 41T^{2} \)
43 \( 1 + 6iT - 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 - 2iT - 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 + 2iT - 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + 8iT - 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−15.48094517667249786984419820405, −14.35357484071792060463199967707, −13.13503890469136268151357292289, −11.32139467924123906337733892314, −10.43059520603588603966913018618, −9.641698746713342675298466711552, −8.451466216636959955255634250710, −7.07329553608668792226551336916, −4.93630899930433513855533732459, −3.60296988112209500923280590170, 1.18396783807390036137058712068, 4.33689139383410961559481614521, 6.26448749475629147424939311323, 7.955940430465568710907714334923, 8.403686816275583015400740968023, 9.764858650237680717334496654142, 11.35492077341153365777874748609, 12.58532070036526148632685119076, 13.29225623876548200001606745781, 14.30599507841215133090697000144

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