Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $-0.312 - 0.949i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.99i·2-s − 1.58·3-s − 1.97·4-s − 2.66i·5-s + 3.16i·6-s + 1.73i·7-s − 0.0439i·8-s − 0.476·9-s − 5.32·10-s − 2.93i·11-s + 3.14·12-s + (−3.42 + 1.12i)13-s + 3.46·14-s + 4.23i·15-s − 4.04·16-s − 1.90·17-s + ⋯
L(s)  = 1  − 1.41i·2-s − 0.917·3-s − 0.988·4-s − 1.19i·5-s + 1.29i·6-s + 0.657i·7-s − 0.0155i·8-s − 0.158·9-s − 1.68·10-s − 0.886i·11-s + 0.906·12-s + (−0.949 + 0.312i)13-s + 0.927·14-s + 1.09i·15-s − 1.01·16-s − 0.461·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.312 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.312 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $-0.312 - 0.949i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (311, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 403,\ (\ :1/2),\ -0.312 - 0.949i)$
$L(1)$  $\approx$  $0.266370 + 0.368015i$
$L(\frac12)$  $\approx$  $0.266370 + 0.368015i$
$L(\frac{3}{2})$   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 \( 1 + (3.42 - 1.12i)T \)
31 \( 1 + iT \)
good2 \( 1 + 1.99iT - 2T^{2} \)
3 \( 1 + 1.58T + 3T^{2} \)
5 \( 1 + 2.66iT - 5T^{2} \)
7 \( 1 - 1.73iT - 7T^{2} \)
11 \( 1 + 2.93iT - 11T^{2} \)
17 \( 1 + 1.90T + 17T^{2} \)
19 \( 1 - 4.24iT - 19T^{2} \)
23 \( 1 - 3.75T + 23T^{2} \)
29 \( 1 + 4.27T + 29T^{2} \)
37 \( 1 + 3.05iT - 37T^{2} \)
41 \( 1 - 5.80iT - 41T^{2} \)
43 \( 1 + 11.6T + 43T^{2} \)
47 \( 1 + 10.5iT - 47T^{2} \)
53 \( 1 + 13.8T + 53T^{2} \)
59 \( 1 + 13.7iT - 59T^{2} \)
61 \( 1 - 5.05T + 61T^{2} \)
67 \( 1 + 12.7iT - 67T^{2} \)
71 \( 1 + 5.83iT - 71T^{2} \)
73 \( 1 - 4.92iT - 73T^{2} \)
79 \( 1 - 0.743T + 79T^{2} \)
83 \( 1 - 0.165iT - 83T^{2} \)
89 \( 1 + 1.85iT - 89T^{2} \)
97 \( 1 + 6.35iT - 97T^{2} \)
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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

−11.00038108242604399547689877696, −9.860012887954088825720643986430, −9.084616149914109913593540343141, −8.270540120756451826735969149257, −6.57176890388937031872584393033, −5.42073066789857775087095745676, −4.72100358947154088691213871683, −3.29769303978150257956464635104, −1.83376102809879966433373630506, −0.31724277709481984017687016418, 2.72346539437469374169943870977, 4.57731353223789115205886834875, 5.37405724872223766726919824907, 6.56223410730717966446801649104, 6.98434077165417412341121242272, 7.65940193607962270678157146362, 9.019907685991616334129645787671, 10.19666222417065809329737636702, 10.97627537415658053162310905441, 11.65831476272166502655644411169

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