Properties

Degree 2
Conductor $ 5 \cdot 13 $
Sign $0.868 - 0.496i$
Motivic weight 1
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

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(65\)    =    \(5 \cdot 13\)
\( \varepsilon \)  =  $0.868 - 0.496i$
motivic weight  =  \(1\)
character  :  $\chi_{65} (64, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 65,\ (\ :1/2),\ 0.868 - 0.496i)$
$L(1)$  $\approx$  $1.06832 + 0.283707i$
$L(\frac12)$  $\approx$  $1.06832 + 0.283707i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;13\}$, \(F_p(T)\) is a polynomial of degree 2. If $p \in \{5,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 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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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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