Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $-0.360 + 0.932i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.40i·2-s + 1.52·3-s + 0.0258·4-s − 1.01i·5-s − 2.14i·6-s − 2.90i·7-s − 2.84i·8-s − 0.662·9-s − 1.42·10-s + 2.94i·11-s + 0.0394·12-s + (3.36 + 1.30i)13-s − 4.08·14-s − 1.55i·15-s − 3.94·16-s − 5.52·17-s + ⋯
L(s)  = 1  − 0.993i·2-s + 0.882·3-s + 0.0129·4-s − 0.454i·5-s − 0.876i·6-s − 1.09i·7-s − 1.00i·8-s − 0.220·9-s − 0.451·10-s + 0.886i·11-s + 0.0113·12-s + (0.932 + 0.360i)13-s − 1.09·14-s − 0.401i·15-s − 0.986·16-s − 1.34·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.360 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.360 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $-0.360 + 0.932i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (311, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 403,\ (\ :1/2),\ -0.360 + 0.932i)$
$L(1)$  $\approx$  $1.09040 - 1.59125i$
$L(\frac12)$  $\approx$  $1.09040 - 1.59125i$
$L(\frac{3}{2})$   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 \( 1 + (-3.36 - 1.30i)T \)
31 \( 1 + iT \)
good2 \( 1 + 1.40iT - 2T^{2} \)
3 \( 1 - 1.52T + 3T^{2} \)
5 \( 1 + 1.01iT - 5T^{2} \)
7 \( 1 + 2.90iT - 7T^{2} \)
11 \( 1 - 2.94iT - 11T^{2} \)
17 \( 1 + 5.52T + 17T^{2} \)
19 \( 1 - 3.13iT - 19T^{2} \)
23 \( 1 - 3.87T + 23T^{2} \)
29 \( 1 - 5.59T + 29T^{2} \)
37 \( 1 + 3.92iT - 37T^{2} \)
41 \( 1 - 12.1iT - 41T^{2} \)
43 \( 1 - 3.52T + 43T^{2} \)
47 \( 1 + 9.64iT - 47T^{2} \)
53 \( 1 + 11.3T + 53T^{2} \)
59 \( 1 - 12.4iT - 59T^{2} \)
61 \( 1 - 5.57T + 61T^{2} \)
67 \( 1 - 4.26iT - 67T^{2} \)
71 \( 1 - 9.44iT - 71T^{2} \)
73 \( 1 - 0.784iT - 73T^{2} \)
79 \( 1 - 6.11T + 79T^{2} \)
83 \( 1 + 8.81iT - 83T^{2} \)
89 \( 1 + 6.58iT - 89T^{2} \)
97 \( 1 - 6.44iT - 97T^{2} \)
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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

−10.96719922660909558730485103411, −10.19630531460906907894785218633, −9.249155707450736005823927614318, −8.476851051303428542349849323411, −7.31107116119103418018356402112, −6.44417331231810463384111130614, −4.55511599449793335689977169977, −3.74997442813074076189156088676, −2.57064572228899338820297703333, −1.30118858579066313038652041047, 2.42154079087991883993525655650, 3.17389936341910389230158508531, 5.03287782623007369850227860210, 6.09308113057033159910577014838, 6.73130612450714771852296361553, 7.995388277612001790923893620086, 8.729446771879556977563628383519, 9.056185674687108859323307923448, 10.98553844829937722371593772297, 11.17474368771168653871063246290

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