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  :  $\chi_{731} (214, \cdot )$
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

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

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