Properties

Degree 2
Conductor 13
Sign $-0.148 + 0.988i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.74 + 10.2i)2-s + (−22.3 − 38.7i)3-s + (−41.8 − 24.1i)4-s + (−35.0 − 35.0i)5-s + (458. − 122. i)6-s + (−112. − 420. i)7-s + (−117. + 117. i)8-s + (−638. + 1.10e3i)9-s + (454. − 262. i)10-s + (−1.83e3 − 492. i)11-s + 2.16e3i·12-s + (2.15e3 + 410. i)13-s + 4.61e3·14-s + (−574. + 2.14e3i)15-s + (−2.42e3 − 4.20e3i)16-s + (2.97e3 + 1.71e3i)17-s + ⋯
L(s)  = 1  + (−0.342 + 1.27i)2-s + (−0.829 − 1.43i)3-s + (−0.653 − 0.377i)4-s + (−0.280 − 0.280i)5-s + (2.12 − 0.568i)6-s + (−0.328 − 1.22i)7-s + (−0.229 + 0.229i)8-s + (−0.876 + 1.51i)9-s + (0.454 − 0.262i)10-s + (−1.38 − 0.369i)11-s + 1.25i·12-s + (0.982 + 0.186i)13-s + 1.68·14-s + (−0.170 + 0.634i)15-s + (−0.592 − 1.02i)16-s + (0.605 + 0.349i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.148 + 0.988i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.148 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(13\)
\( \varepsilon \)  =  $-0.148 + 0.988i$
motivic weight  =  \(6\)
character  :  $\chi_{13} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 13,\ (\ :3),\ -0.148 + 0.988i)$
$L(\frac{7}{2})$  $\approx$  $0.304056 - 0.353166i$
$L(\frac12)$  $\approx$  $0.304056 - 0.353166i$
$L(4)$   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 \( 1 + (-2.15e3 - 410. i)T \)
good2 \( 1 + (2.74 - 10.2i)T + (-55.4 - 32i)T^{2} \)
3 \( 1 + (22.3 + 38.7i)T + (-364.5 + 631. i)T^{2} \)
5 \( 1 + (35.0 + 35.0i)T + 1.56e4iT^{2} \)
7 \( 1 + (112. + 420. i)T + (-1.01e5 + 5.88e4i)T^{2} \)
11 \( 1 + (1.83e3 + 492. i)T + (1.53e6 + 8.85e5i)T^{2} \)
17 \( 1 + (-2.97e3 - 1.71e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (-1.31e3 + 353. i)T + (4.07e7 - 2.35e7i)T^{2} \)
23 \( 1 + (-4.73e3 + 2.73e3i)T + (7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (-6.24e3 - 1.08e4i)T + (-2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + (3.93e4 + 3.93e4i)T + 8.87e8iT^{2} \)
37 \( 1 + (4.16e4 + 1.11e4i)T + (2.22e9 + 1.28e9i)T^{2} \)
41 \( 1 + (-9.28e3 + 3.46e4i)T + (-4.11e9 - 2.37e9i)T^{2} \)
43 \( 1 + (-8.07e4 - 4.66e4i)T + (3.16e9 + 5.47e9i)T^{2} \)
47 \( 1 + (-3.16e4 + 3.16e4i)T - 1.07e10iT^{2} \)
53 \( 1 + 1.62e5T + 2.21e10T^{2} \)
59 \( 1 + (-1.51e4 - 5.64e4i)T + (-3.65e10 + 2.10e10i)T^{2} \)
61 \( 1 + (-9.81e4 + 1.70e5i)T + (-2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (3.11e4 - 1.16e5i)T + (-7.83e10 - 4.52e10i)T^{2} \)
71 \( 1 + (-4.72e4 + 1.26e4i)T + (1.10e11 - 6.40e10i)T^{2} \)
73 \( 1 + (-1.89e4 + 1.89e4i)T - 1.51e11iT^{2} \)
79 \( 1 + 9.10e4T + 2.43e11T^{2} \)
83 \( 1 + (-2.40e5 - 2.40e5i)T + 3.26e11iT^{2} \)
89 \( 1 + (1.15e6 + 3.10e5i)T + (4.30e11 + 2.48e11i)T^{2} \)
97 \( 1 + (-4.69e5 + 1.25e5i)T + (7.21e11 - 4.16e11i)T^{2} \)
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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

−17.95334278669009648480816954325, −16.82030646816184086152179064344, −16.02344422416308575376306776280, −13.87932205015604590121520768425, −12.74699061973102648556330287426, −10.96705166813043647160099514372, −8.093241306291882742182780888206, −7.13588796173315896874779311629, −5.78132330457806841212759422896, −0.43183964991025141328242473483, 3.22527062811828659237485880773, 5.48795353621795341263773439617, 9.157717977757484835211837689319, 10.38767000764850213085276152642, 11.29246482914916893539678452389, 12.54910472683845283680144181444, 15.37522226127315051786568723449, 15.94387847811095746641095494988, 17.95866244083797625393955362505, 18.91273740598438607334965106424

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