Properties

Label 2-13-13.6-c2-0-0
Degree $2$
Conductor $13$
Sign $0.846 + 0.533i$
Analytic cond. $0.354224$
Root an. cond. $0.595167$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 1.86i)2-s + (−1.36 + 2.36i)3-s + (0.232 − 0.133i)4-s + (−4.36 + 4.36i)5-s + (5.09 + 1.36i)6-s + (2.26 − 8.46i)7-s + (−5.83 − 5.83i)8-s + (0.767 + 1.33i)9-s + (10.3 + 5.96i)10-s + (6.19 − 1.66i)11-s + 0.732i·12-s + (−6.5 + 11.2i)13-s − 16.9·14-s + (−4.36 − 16.2i)15-s + (−7.42 + 12.8i)16-s + (9.99 − 5.76i)17-s + ⋯
L(s)  = 1  + (−0.250 − 0.933i)2-s + (−0.455 + 0.788i)3-s + (0.0580 − 0.0334i)4-s + (−0.873 + 0.873i)5-s + (0.849 + 0.227i)6-s + (0.323 − 1.20i)7-s + (−0.728 − 0.728i)8-s + (0.0853 + 0.147i)9-s + (1.03 + 0.596i)10-s + (0.563 − 0.150i)11-s + 0.0610i·12-s + (−0.5 + 0.866i)13-s − 1.20·14-s + (−0.291 − 1.08i)15-s + (−0.464 + 0.804i)16-s + (0.587 − 0.339i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 + 0.533i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.846 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $0.846 + 0.533i$
Analytic conductor: \(0.354224\)
Root analytic conductor: \(0.595167\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :1),\ 0.846 + 0.533i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.586286 - 0.169291i\)
\(L(\frac12)\) \(\approx\) \(0.586286 - 0.169291i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (6.5 - 11.2i)T \)
good2 \( 1 + (0.5 + 1.86i)T + (-3.46 + 2i)T^{2} \)
3 \( 1 + (1.36 - 2.36i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (4.36 - 4.36i)T - 25iT^{2} \)
7 \( 1 + (-2.26 + 8.46i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (-6.19 + 1.66i)T + (104. - 60.5i)T^{2} \)
17 \( 1 + (-9.99 + 5.76i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-3.36 - 0.901i)T + (312. + 180.5i)T^{2} \)
23 \( 1 + (8.49 + 4.90i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-5.69 + 9.86i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-1.92 + 1.92i)T - 961iT^{2} \)
37 \( 1 + (42.1 - 11.2i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (-5.08 - 18.9i)T + (-1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (-45 + 25.9i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-0.320 - 0.320i)T + 2.20e3iT^{2} \)
53 \( 1 - 78.7T + 2.80e3T^{2} \)
59 \( 1 + (-10.9 + 40.9i)T + (-3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (49.1 + 85.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-19.9 - 74.5i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (31.0 + 8.31i)T + (4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (-48.2 - 48.2i)T + 5.32e3iT^{2} \)
79 \( 1 + 82.7T + 6.24e3T^{2} \)
83 \( 1 + (69.5 - 69.5i)T - 6.88e3iT^{2} \)
89 \( 1 + (31.8 - 8.52i)T + (6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (-74.8 - 20.0i)T + (8.14e3 + 4.70e3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.67380957328236826127553915476, −18.79531553297321548176777089468, −16.93927975372165790455416367734, −15.69349632537767292893142502756, −14.23948615269701998674708152208, −11.79909410586806601335509550978, −10.93174628761151014090250123475, −9.949203577202307198335670126866, −7.14798786949140281813477201047, −3.95683416420677932391690554448, 5.69553781794938833798720680035, 7.49426845231199612039305488387, 8.748529320555201240166277877670, 11.94096394180814851765994738485, 12.35508748671146947038322297182, 14.92450749565529743467736844708, 15.88555550706980427605586348646, 17.22051251081396437134079460275, 18.20982450485576413706454723373, 19.69946131770991524875045309811

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