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  :  $\chi_{13} (6, \cdot )$
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

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

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