Properties

Degree 2
Conductor 13
Sign $0.914 + 0.404i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (12.5 − 3.35i)2-s + (−1.31 + 2.27i)3-s + (89.8 − 51.8i)4-s + (24.9 + 24.9i)5-s + (−8.79 + 32.8i)6-s + (−400. − 107. i)7-s + (363. − 363. i)8-s + (361. + 625. i)9-s + (395. + 228. i)10-s + (−169. − 631. i)11-s + 272. i·12-s + (−1.80e3 + 1.24e3i)13-s − 5.36e3·14-s + (−89.4 + 23.9i)15-s + (10.7 − 18.6i)16-s + (4.04e3 − 2.33e3i)17-s + ⋯
L(s)  = 1  + (1.56 − 0.418i)2-s + (−0.0486 + 0.0841i)3-s + (1.40 − 0.810i)4-s + (0.199 + 0.199i)5-s + (−0.0407 + 0.151i)6-s + (−1.16 − 0.312i)7-s + (0.710 − 0.710i)8-s + (0.495 + 0.857i)9-s + (0.395 + 0.228i)10-s + (−0.127 − 0.474i)11-s + 0.157i·12-s + (−0.822 + 0.568i)13-s − 1.95·14-s + (−0.0264 + 0.00710i)15-s + (0.00262 − 0.00454i)16-s + (0.823 − 0.475i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(13\)
\( \varepsilon \)  =  $0.914 + 0.404i$
motivic weight  =  \(6\)
character  :  $\chi_{13} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 13,\ (\ :3),\ 0.914 + 0.404i)$
$L(\frac{7}{2})$  $\approx$  $2.63579 - 0.557528i$
$L(\frac12)$  $\approx$  $2.63579 - 0.557528i$
$L(4)$   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 \( 1 + (1.80e3 - 1.24e3i)T \)
good2 \( 1 + (-12.5 + 3.35i)T + (55.4 - 32i)T^{2} \)
3 \( 1 + (1.31 - 2.27i)T + (-364.5 - 631. i)T^{2} \)
5 \( 1 + (-24.9 - 24.9i)T + 1.56e4iT^{2} \)
7 \( 1 + (400. + 107. i)T + (1.01e5 + 5.88e4i)T^{2} \)
11 \( 1 + (169. + 631. i)T + (-1.53e6 + 8.85e5i)T^{2} \)
17 \( 1 + (-4.04e3 + 2.33e3i)T + (1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (-2.58e3 + 9.66e3i)T + (-4.07e7 - 2.35e7i)T^{2} \)
23 \( 1 + (-1.38e4 - 7.98e3i)T + (7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + (-6.61e3 + 1.14e4i)T + (-2.97e8 - 5.15e8i)T^{2} \)
31 \( 1 + (6.05e3 + 6.05e3i)T + 8.87e8iT^{2} \)
37 \( 1 + (-2.10e4 - 7.84e4i)T + (-2.22e9 + 1.28e9i)T^{2} \)
41 \( 1 + (1.14e5 - 3.06e4i)T + (4.11e9 - 2.37e9i)T^{2} \)
43 \( 1 + (3.86e4 - 2.22e4i)T + (3.16e9 - 5.47e9i)T^{2} \)
47 \( 1 + (-3.40e4 + 3.40e4i)T - 1.07e10iT^{2} \)
53 \( 1 + 1.46e4T + 2.21e10T^{2} \)
59 \( 1 + (1.01e5 + 2.73e4i)T + (3.65e10 + 2.10e10i)T^{2} \)
61 \( 1 + (-1.07e5 - 1.85e5i)T + (-2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (-4.70e5 + 1.26e5i)T + (7.83e10 - 4.52e10i)T^{2} \)
71 \( 1 + (-4.93e3 + 1.84e4i)T + (-1.10e11 - 6.40e10i)T^{2} \)
73 \( 1 + (-1.58e5 + 1.58e5i)T - 1.51e11iT^{2} \)
79 \( 1 + 2.80e5T + 2.43e11T^{2} \)
83 \( 1 + (6.62e5 + 6.62e5i)T + 3.26e11iT^{2} \)
89 \( 1 + (4.80e3 + 1.79e4i)T + (-4.30e11 + 2.48e11i)T^{2} \)
97 \( 1 + (2.95e5 - 1.10e6i)T + (-7.21e11 - 4.16e11i)T^{2} \)
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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

−18.97989700792454693046813824439, −16.67612795635632479986720840853, −15.42813979117905135824408865485, −13.82878042101889397414160075888, −13.10283314819927842766105928386, −11.57123327852619920052434813203, −9.974427987045679578031678802172, −6.80820595220534986423685446321, −4.93376219448312407065543485732, −2.94658798675343089936114800787, 3.43159906759084694609680032827, 5.52615242125628885299098096403, 6.98637086437242340352945412564, 9.775081336179492267748737288034, 12.44842132678913635485404775946, 12.72786104119378318083123562195, 14.52867145562776349338401447099, 15.51810360484385998754452753431, 16.81469291003899860284901308174, 18.72688913869080207417316232146

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