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Properties

Label 2-65-65.47-c1-0-2
Degree $2$
Conductor $65$
Sign $0.927 - 0.374i$
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  + i·2-s + (1 − i)3-s + 4-s + (−2 − i)5-s + (1 + i)6-s − 2·7-s + 3i·8-s + i·9-s + (1 − 2i)10-s + (−1 + i)11-s + (1 − i)12-s + (−2 − 3i)13-s − 2i·14-s + (−3 + i)15-s − 16-s + (−1 + i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.577 − 0.577i)3-s + 0.5·4-s + (−0.894 − 0.447i)5-s + (0.408 + 0.408i)6-s − 0.755·7-s + 1.06i·8-s + 0.333i·9-s + (0.316 − 0.632i)10-s + (−0.301 + 0.301i)11-s + (0.288 − 0.288i)12-s + (−0.554 − 0.832i)13-s − 0.534i·14-s + (−0.774 + 0.258i)15-s − 0.250·16-s + (−0.242 + 0.242i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.927 - 0.374i$
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: \((2,\ 65,\ (\ :1/2),\ 0.927 - 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.979091 + 0.190293i\)
\(L(\frac12)\) \(\approx\) \(0.979091 + 0.190293i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2 + i)T \)
13 \( 1 + (2 + 3i)T \)
good2 \( 1 - iT - 2T^{2} \)
3 \( 1 + (-1 + i)T - 3iT^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 + (1 - i)T - 11iT^{2} \)
17 \( 1 + (1 - i)T - 17iT^{2} \)
19 \( 1 + (-5 + 5i)T - 19iT^{2} \)
23 \( 1 + (3 + 3i)T + 23iT^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (-5 - 5i)T + 31iT^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + (7 + 7i)T + 41iT^{2} \)
43 \( 1 + (-1 - i)T + 43iT^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + (-5 + 5i)T - 53iT^{2} \)
59 \( 1 + (-7 - 7i)T + 59iT^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + (-1 - i)T + 71iT^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 + 2iT - 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (5 + 5i)T + 89iT^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−15.28579776964001605097298587298, −13.95532139868130104566786074623, −12.83582081536284619655210794362, −11.87807583563561442245591663179, −10.44727603384268724981891056381, −8.672039690771289654349266605249, −7.69350949572141222845109232994, −6.91714408521196356372826667724, −5.13100718728728937839185291943, −2.82409610158548274010114456850, 2.96181350563692654476995942473, 3.93587120510712779749531006049, 6.47283955809375816477191327139, 7.75886313353359764562170939638, 9.463473378377305064427803714975, 10.22778443578192177665180726873, 11.62223127762783070483086381618, 12.19842745299681660905524811285, 13.78404864990385496336077884859, 15.01761079605975656508322141111

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