Properties

Degree 2
Conductor 13
Sign $0.964 - 0.265i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 0.866i)2-s + (−1 − 1.73i)3-s + (0.5 − 0.866i)4-s + 1.73i·5-s + (3 + 1.73i)6-s − 1.73i·8-s + (−0.499 + 0.866i)9-s + (−1.49 − 2.59i)10-s − 2·12-s + (−2.5 + 2.59i)13-s + (2.99 − 1.73i)15-s + (2.49 + 4.33i)16-s + (1.5 − 2.59i)17-s − 1.73i·18-s + (−3 − 1.73i)19-s + (1.50 + 0.866i)20-s + ⋯
L(s)  = 1  + (−1.06 + 0.612i)2-s + (−0.577 − 0.999i)3-s + (0.250 − 0.433i)4-s + 0.774i·5-s + (1.22 + 0.707i)6-s − 0.612i·8-s + (−0.166 + 0.288i)9-s + (−0.474 − 0.821i)10-s − 0.577·12-s + (−0.693 + 0.720i)13-s + (0.774 − 0.447i)15-s + (0.624 + 1.08i)16-s + (0.363 − 0.630i)17-s − 0.408i·18-s + (−0.688 − 0.397i)19-s + (0.335 + 0.193i)20-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(13\)
\( \varepsilon \)  =  $0.964 - 0.265i$
motivic weight  =  \(1\)
character  :  $\chi_{13} (10, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 13,\ (\ :1/2),\ 0.964 - 0.265i)$
$L(1)$  $\approx$  $0.298115 + 0.0402203i$
$L(\frac12)$  $\approx$  $0.298115 + 0.0402203i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 13$, \(F_p\) is a polynomial of degree 2. If $p = 13$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad13 \( 1 + (2.5 - 2.59i)T \)
good2 \( 1 + (1.5 - 0.866i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (1 + 1.73i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 1.73iT - 5T^{2} \)
7 \( 1 + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 + 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + (-7.5 + 4.33i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.5 - 2.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.46iT - 47T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 + (-6 - 3.46i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3 + 1.73i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 - 1.73i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.73iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 13.8iT - 83T^{2} \)
89 \( 1 + (6 - 3.46i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6 - 3.46i)T + (48.5 + 84.0i)T^{2} \)
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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

−19.33846444803067811938292685952, −18.56576970884593164181156978722, −17.65895984226676047880460224981, −16.64032221486140008944897783369, −14.95140792841582142043623560609, −13.06339550571964159220878092128, −11.45861743544082710207892638661, −9.565832078364084782726879302558, −7.54467569516254146308765020154, −6.57037775431560330867930744613, 5.06823463540602925038855173519, 8.455526054396895167012626272507, 9.917477007979000496132305323081, 10.84883986345867375872722724058, 12.51224222861724419404676510427, 14.90357896669741344337333157041, 16.58357118750575786137645040830, 17.20457702255851851831821238873, 18.76925999761091455577670097918, 20.10703301806788866096607442711

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