Properties

Degree 2
Conductor $ 5 \cdot 13 $
Sign $0.979 + 0.202i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.202i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.202i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(65\)    =    \(5 \cdot 13\)
\( \varepsilon \)  =  $0.979 + 0.202i$
motivic weight  =  \(1\)
character  :  $\chi_{65} (57, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 65,\ (\ :1/2),\ 0.979 + 0.202i)$
$L(1)$  $\approx$  $1.15677 - 0.118445i$
$L(\frac12)$  $\approx$  $1.15677 - 0.118445i$
$L(\frac{3}{2})$   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 \( 1 + (-1 + i)T - 3iT^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
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 \( 1 - 6iT - 47T^{2} \)
53 \( 1 + (-5 + 5i)T - 53iT^{2} \)
59 \( 1 + (7 - 7i)T - 59iT^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + (-1 + i)T - 71iT^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 + 2iT - 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + (-5 + 5i)T - 89iT^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−14.57493038135502634456586473826, −13.52274998356247544693642718630, −13.31812493799828575988047982210, −11.62722793832416919113155724919, −10.54005650859245523922722823536, −8.853818757476344028243810099807, −7.66232163139685793882688117314, −6.41972131090882448749640525167, −4.46410220828931050158684313703, −3.03470510595937031587764894302, 3.44443955823377383091885803600, 4.62117467209420483661964835873, 5.93277188311989188901804805163, 8.436669596736453868347650327525, 8.841673258484196603041226504005, 10.24348985741680663624456227474, 12.16067230501220492707175245836, 12.65265466309089736042431370074, 13.87875728445781756249227402376, 15.14223922415733873819639039462

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