Properties

Degree 2
Conductor 13
Sign $-0.598 + 0.801i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.76 − 6.59i)2-s + (−10.7 − 18.6i)3-s + (15.0 + 8.70i)4-s + (−144. − 144. i)5-s + (−141. + 37.9i)6-s + (49.5 + 184. i)7-s + (392. − 392. i)8-s + (133. − 230. i)9-s + (−1.21e3 + 699. i)10-s + (1.43e3 + 385. i)11-s − 374. i·12-s + (1.70e3 + 1.38e3i)13-s + 1.30e3·14-s + (−1.14e3 + 4.25e3i)15-s + (−1.33e3 − 2.32e3i)16-s + (−5.32e3 − 3.07e3i)17-s + ⋯
L(s)  = 1  + (0.220 − 0.824i)2-s + (−0.398 − 0.689i)3-s + (0.235 + 0.135i)4-s + (−1.15 − 1.15i)5-s + (−0.656 + 0.175i)6-s + (0.144 + 0.539i)7-s + (0.767 − 0.767i)8-s + (0.182 − 0.316i)9-s + (−1.21 + 0.699i)10-s + (1.08 + 0.289i)11-s − 0.216i·12-s + (0.777 + 0.628i)13-s + 0.476·14-s + (−0.337 + 1.26i)15-s + (−0.327 − 0.566i)16-s + (−1.08 − 0.625i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.598 + 0.801i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.598 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(13\)
\( \varepsilon \)  =  $-0.598 + 0.801i$
motivic weight  =  \(6\)
character  :  $\chi_{13} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 13,\ (\ :3),\ -0.598 + 0.801i)$
$L(\frac{7}{2})$  $\approx$  $0.603979 - 1.20506i$
$L(\frac12)$  $\approx$  $0.603979 - 1.20506i$
$L(4)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 13$,\(F_p(T)\) is a polynomial of degree 2. If $p = 13$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad13 \( 1 + (-1.70e3 - 1.38e3i)T \)
good2 \( 1 + (-1.76 + 6.59i)T + (-55.4 - 32i)T^{2} \)
3 \( 1 + (10.7 + 18.6i)T + (-364.5 + 631. i)T^{2} \)
5 \( 1 + (144. + 144. i)T + 1.56e4iT^{2} \)
7 \( 1 + (-49.5 - 184. i)T + (-1.01e5 + 5.88e4i)T^{2} \)
11 \( 1 + (-1.43e3 - 385. i)T + (1.53e6 + 8.85e5i)T^{2} \)
17 \( 1 + (5.32e3 + 3.07e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (-2.66e3 + 715. i)T + (4.07e7 - 2.35e7i)T^{2} \)
23 \( 1 + (1.10e3 - 635. i)T + (7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (-1.43e4 - 2.48e4i)T + (-2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + (2.35e3 + 2.35e3i)T + 8.87e8iT^{2} \)
37 \( 1 + (-3.75e4 - 1.00e4i)T + (2.22e9 + 1.28e9i)T^{2} \)
41 \( 1 + (1.83e4 - 6.86e4i)T + (-4.11e9 - 2.37e9i)T^{2} \)
43 \( 1 + (5.18e4 + 2.99e4i)T + (3.16e9 + 5.47e9i)T^{2} \)
47 \( 1 + (-1.37e5 + 1.37e5i)T - 1.07e10iT^{2} \)
53 \( 1 + 1.12e5T + 2.21e10T^{2} \)
59 \( 1 + (-9.04e4 - 3.37e5i)T + (-3.65e10 + 2.10e10i)T^{2} \)
61 \( 1 + (8.32e4 - 1.44e5i)T + (-2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (4.23e4 - 1.57e5i)T + (-7.83e10 - 4.52e10i)T^{2} \)
71 \( 1 + (-3.12e5 + 8.36e4i)T + (1.10e11 - 6.40e10i)T^{2} \)
73 \( 1 + (4.13e5 - 4.13e5i)T - 1.51e11iT^{2} \)
79 \( 1 - 4.24e4T + 2.43e11T^{2} \)
83 \( 1 + (4.32e5 + 4.32e5i)T + 3.26e11iT^{2} \)
89 \( 1 + (1.12e5 + 3.00e4i)T + (4.30e11 + 2.48e11i)T^{2} \)
97 \( 1 + (-9.66e5 + 2.59e5i)T + (7.21e11 - 4.16e11i)T^{2} \)
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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

−18.38702253600196603797683645791, −16.67026779092733180486501732957, −15.56563331678616367245034732680, −13.16338159410531502730459430833, −11.94773036045882689411083353738, −11.62778483172381174189909809653, −8.904881709634196516532833754763, −6.97103952461439090808595208413, −4.16058267109963425699917147670, −1.21494263716344480031016721152, 4.10537533539872181637112472632, 6.39895622897823995875090093215, 7.81018889946144284810049006180, 10.66145551959665267844570672066, 11.31073204562829861997412390875, 14.01039294765879397788198638255, 15.28545834484763289047757335732, 15.92358186560318172590479485811, 17.27037346570438712146868998767, 19.12820171426778694329669805007

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