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Properties

Label 2-65-65.64-c1-0-3
Degree $2$
Conductor $65$
Sign $0.868 + 0.496i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.496i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.868 + 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.868 + 0.496i$
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: \((2,\ 65,\ (\ :1/2),\ 0.868 + 0.496i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06832 - 0.283707i\)
\(L(\frac12)\) \(\approx\) \(1.06832 - 0.283707i\)
\(L(\frac{3}{2})\) 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

−14.34862516746341784522932702950, −13.78823751106707503594314175371, −12.71714119571974731062634462663, −11.98239077020634614152784228637, −10.33586677104772337841262224795, −8.972634087180848766990541926844, −7.23591734839184107054042839302, −6.42372356969681330659811031444, −4.72173132163098801836255991991, −2.58431702142523155261267406595, 3.64189765661446866968181750927, 4.88156617889206626562591184346, 5.73514302994749178233236282911, 8.260808297600080955067953336526, 9.463934432947145288101362720902, 10.15382394013077472054980476610, 11.90311223456943156683204875337, 12.93251967907106945395356216934, 13.92357669379673350762503868502, 14.93053022598393681489846106157

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