Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $-0.913 + 0.407i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.01i·2-s − 1.05·3-s + 0.974·4-s − 3.24i·5-s + 1.06i·6-s − 1.79i·7-s − 3.01i·8-s − 1.89·9-s − 3.28·10-s + 6.09i·11-s − 1.02·12-s + (−1.47 − 3.29i)13-s − 1.81·14-s + 3.41i·15-s − 1.09·16-s + 4.79·17-s + ⋯
L(s)  = 1  − 0.715i·2-s − 0.607·3-s + 0.487·4-s − 1.45i·5-s + 0.434i·6-s − 0.678i·7-s − 1.06i·8-s − 0.631·9-s − 1.03·10-s + 1.83i·11-s − 0.295·12-s + (−0.407 − 0.913i)13-s − 0.485·14-s + 0.880i·15-s − 0.274·16-s + 1.16·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 + 0.407i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.913 + 0.407i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $-0.913 + 0.407i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (311, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 403,\ (\ :1/2),\ -0.913 + 0.407i)$
$L(1)$  $\approx$  $0.227580 - 1.06753i$
$L(\frac12)$  $\approx$  $0.227580 - 1.06753i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{13,\;31\}$, \(F_p(T)\) is a polynomial of degree 2. If $p \in \{13,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad13 \( 1 + (1.47 + 3.29i)T \)
31 \( 1 - iT \)
good2 \( 1 + 1.01iT - 2T^{2} \)
3 \( 1 + 1.05T + 3T^{2} \)
5 \( 1 + 3.24iT - 5T^{2} \)
7 \( 1 + 1.79iT - 7T^{2} \)
11 \( 1 - 6.09iT - 11T^{2} \)
17 \( 1 - 4.79T + 17T^{2} \)
19 \( 1 + 1.84iT - 19T^{2} \)
23 \( 1 + 7.94T + 23T^{2} \)
29 \( 1 + 2.23T + 29T^{2} \)
37 \( 1 + 8.64iT - 37T^{2} \)
41 \( 1 + 0.0498iT - 41T^{2} \)
43 \( 1 - 4.76T + 43T^{2} \)
47 \( 1 - 4.00iT - 47T^{2} \)
53 \( 1 - 0.588T + 53T^{2} \)
59 \( 1 + 3.49iT - 59T^{2} \)
61 \( 1 - 12.1T + 61T^{2} \)
67 \( 1 - 1.72iT - 67T^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + 2.28iT - 73T^{2} \)
79 \( 1 - 16.1T + 79T^{2} \)
83 \( 1 - 10.2iT - 83T^{2} \)
89 \( 1 - 7.99iT - 89T^{2} \)
97 \( 1 + 10.1iT - 97T^{2} \)
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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

−10.88270909596552157995205826476, −10.06435272926643452910069214099, −9.436846594518341254975733016875, −7.983215653127614643728002352023, −7.24371390787889943306604764998, −5.85379828833843473533549463363, −4.94113793092030304031674114951, −3.87359674124128814525397028243, −2.16197208797998745045430217266, −0.74302558891654016651960838591, 2.43201092439944989218609452305, 3.42914439368563623582348716859, 5.57231995976565159605084947453, 5.98443139406144540474596435931, 6.69753712817983710491653570656, 7.82703972707314259283969840703, 8.601394198564366576750697875786, 10.07239716855035433510475632800, 10.92986346737661563086248129139, 11.66790571097378381468142422240

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