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Properties

Degree 2
Conductor $ 3^{2} \cdot 13 $
Sign $0.957 - 0.289i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·4-s + (−1 − i)7-s − i·13-s − 16-s + (−1 + i)19-s + i·25-s + (1 − i)28-s + (1 − i)31-s + (1 + i)37-s + i·49-s + 52-s − i·64-s + (−1 + i)67-s + (−1 − i)73-s + (−1 − i)76-s + ⋯
L(s)  = 1  + i·4-s + (−1 − i)7-s − i·13-s − 16-s + (−1 + i)19-s + i·25-s + (1 − i)28-s + (1 − i)31-s + (1 + i)37-s + i·49-s + 52-s − i·64-s + (−1 + i)67-s + (−1 − i)73-s + (−1 − i)76-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(117\)    =    \(3^{2} \cdot 13\)
\( \varepsilon \)  =  $0.957 - 0.289i$
motivic weight  =  \(0\)
character  :  $\chi_{117} (109, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 117,\ (\ :0),\ 0.957 - 0.289i)$
$L(\frac{1}{2})$  $\approx$  $0.5569436756$
$L(\frac12)$  $\approx$  $0.5569436756$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;13\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{3,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
13 \( 1 + iT \)
good2 \( 1 - iT^{2} \)
5 \( 1 - iT^{2} \)
7 \( 1 + (1 + i)T + iT^{2} \)
11 \( 1 + iT^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (1 - i)T - iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-1 + i)T - iT^{2} \)
37 \( 1 + (-1 - i)T + iT^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + (1 - i)T - iT^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (1 + i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (-1 + i)T - iT^{2} \)
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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

−13.36001136281506947321561544962, −13.05270677275964737188948525381, −11.93499686341665259569750108108, −10.65262963256653539214358760704, −9.701049372459152198528131483308, −8.287424301189132676813928040431, −7.37991611206024721121675636931, −6.18811586118836933892295665132, −4.21585912692895745454493259751, −3.10156635655119368499674695018, 2.45453865405386386796231817165, 4.57945296969333583056588158114, 6.02436325627031227717949276801, 6.74227370788945589021286031122, 8.744582829751551591102352480621, 9.460599077616326652932631890719, 10.52450589542211795910627174439, 11.67036785348349858475122944719, 12.72164914596868163355269831842, 13.80990494819017442544207341531

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