Properties

Degree 2
Conductor $ 17 \cdot 43 $
Sign $0.877 - 0.478i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.41 + 1.41i)4-s + (1.14 − 2.77i)9-s + (1.25 − 1.87i)11-s + (1.77 + 1.77i)13-s − 4.00i·16-s + (1.05 + 3.98i)17-s + (1.81 + 1.20i)23-s + (4.61 + 1.91i)25-s + (4.54 + 6.80i)31-s + (2.29 + 5.54i)36-s + (0.646 − 0.128i)41-s + (−2.50 + 6.05i)43-s + (0.881 + 4.43i)44-s + (9.65 + 9.65i)47-s + (6.46 − 2.67i)49-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)4-s + (0.382 − 0.923i)9-s + (0.378 − 0.566i)11-s + (0.493 + 0.493i)13-s − 1.00i·16-s + (0.255 + 0.966i)17-s + (0.377 + 0.252i)23-s + (0.923 + 0.382i)25-s + (0.816 + 1.22i)31-s + (0.382 + 0.923i)36-s + (0.100 − 0.0200i)41-s + (−0.382 + 0.923i)43-s + (0.132 + 0.668i)44-s + (1.40 + 1.40i)47-s + (0.923 − 0.382i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.478i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.877 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(731\)    =    \(17 \cdot 43\)
\( \varepsilon \)  =  $0.877 - 0.478i$
motivic weight  =  \(1\)
character  :  $\chi_{731} (386, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 731,\ (\ :1/2),\ 0.877 - 0.478i)$
$L(1)$  $\approx$  $1.33445 + 0.340206i$
$L(\frac12)$  $\approx$  $1.33445 + 0.340206i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{17,\;43\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{17,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad17 \( 1 + (-1.05 - 3.98i)T \)
43 \( 1 + (2.50 - 6.05i)T \)
good2 \( 1 + (1.41 - 1.41i)T^{2} \)
3 \( 1 + (-1.14 + 2.77i)T^{2} \)
5 \( 1 + (-4.61 - 1.91i)T^{2} \)
7 \( 1 + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (-1.25 + 1.87i)T + (-4.20 - 10.1i)T^{2} \)
13 \( 1 + (-1.77 - 1.77i)T + 13iT^{2} \)
19 \( 1 + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-1.81 - 1.20i)T + (8.80 + 21.2i)T^{2} \)
29 \( 1 + (26.7 + 11.0i)T^{2} \)
31 \( 1 + (-4.54 - 6.80i)T + (-11.8 + 28.6i)T^{2} \)
37 \( 1 + (-14.1 + 34.1i)T^{2} \)
41 \( 1 + (-0.646 + 0.128i)T + (37.8 - 15.6i)T^{2} \)
47 \( 1 + (-9.65 - 9.65i)T + 47iT^{2} \)
53 \( 1 + (5.55 + 13.4i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (2.19 + 0.907i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (56.3 - 23.3i)T^{2} \)
67 \( 1 + 0.318iT - 67T^{2} \)
71 \( 1 + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (-9.76 + 14.6i)T + (-30.2 - 72.9i)T^{2} \)
83 \( 1 + (-6.05 + 2.50i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + 89iT^{2} \)
97 \( 1 + (3.29 - 16.5i)T + (-89.6 - 37.1i)T^{2} \)
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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

−10.40868846878331494856580391038, −9.338583343719414413921160687533, −8.841163097743187497989396455366, −8.038978921078566567903978610865, −6.92904537646876789841808412098, −6.14228900557983035870258799583, −4.85229660013148388983106097127, −3.85894418444215360449061594518, −3.16617432962985955057727007375, −1.15092702876879522838142360742, 0.975998848378082664600197881796, 2.46791567754583328729401503250, 4.07648030418838397876149265938, 4.88225587279996787045179685172, 5.68923439234557666924014898304, 6.81568463309991933336935267367, 7.76779833076862585210715050449, 8.735684106720091775943961910662, 9.504238254398560152765831045905, 10.35050015020776566861677161989

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