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Properties

Label 2-65-65.18-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: $\chi_{65} (18, \cdot )$
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

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

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