Properties

Degree 2
Conductor 13
Sign $0.973 + 0.230i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.219 − 0.379i)2-s + (1.84 − 3.19i)3-s + (3.90 + 6.76i)4-s − 17.8·5-s + (−0.807 − 1.39i)6-s + (−2.71 − 4.70i)7-s + 6.93·8-s + (6.71 + 11.6i)9-s + (−3.90 + 6.76i)10-s + (11.2 − 19.4i)11-s + 28.7·12-s + (21.9 − 41.4i)13-s − 2.38·14-s + (−32.8 + 56.8i)15-s + (−29.7 + 51.4i)16-s + (−33.9 − 58.8i)17-s + ⋯
L(s)  = 1  + (0.0775 − 0.134i)2-s + (0.354 − 0.614i)3-s + (0.487 + 0.845i)4-s − 1.59·5-s + (−0.0549 − 0.0951i)6-s + (−0.146 − 0.254i)7-s + 0.306·8-s + (0.248 + 0.430i)9-s + (−0.123 + 0.213i)10-s + (0.307 − 0.532i)11-s + 0.692·12-s + (0.468 − 0.883i)13-s − 0.0455·14-s + (−0.564 + 0.978i)15-s + (−0.464 + 0.804i)16-s + (−0.484 − 0.839i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(13\)
\( \varepsilon \)  =  $0.973 + 0.230i$
motivic weight  =  \(3\)
character  :  $\chi_{13} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 13,\ (\ :3/2),\ 0.973 + 0.230i)$
$L(2)$  $\approx$  $0.982027 - 0.114691i$
$L(\frac12)$  $\approx$  $0.982027 - 0.114691i$
$L(\frac{5}{2})$   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 \( 1 + (-21.9 + 41.4i)T \)
good2 \( 1 + (-0.219 + 0.379i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (-1.84 + 3.19i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + 17.8T + 125T^{2} \)
7 \( 1 + (2.71 + 4.70i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-11.2 + 19.4i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (33.9 + 58.8i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-40.4 - 69.9i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (70.2 - 121. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-53.3 + 92.3i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 276.T + 2.97e4T^{2} \)
37 \( 1 + (-2.14 + 3.71i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (113. - 197. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (13.7 + 23.8i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 318.T + 1.03e5T^{2} \)
53 \( 1 + 67.6T + 1.48e5T^{2} \)
59 \( 1 + (-145. - 252. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (331. + 574. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-212. + 368. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-76.4 - 132. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 117.T + 3.89e5T^{2} \)
79 \( 1 - 202.T + 4.93e5T^{2} \)
83 \( 1 - 336.T + 5.71e5T^{2} \)
89 \( 1 + (359. - 621. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (379. + 657. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−19.67780105452462756970465910990, −18.35101931534678645970805260435, −16.44675986116577346436403731362, −15.60674160626015570317270278982, −13.55961242981874110454866407054, −12.24656201119802978584149472660, −11.12718488914698104803801456386, −8.162302933922337311779364217199, −7.34663469630280678944139989589, −3.58625571942394858965526178874, 4.19022483144951431422246083571, 6.90403086611684217568404125648, 9.009732338948342568151131608751, 10.80163656029455800316775363706, 12.14658068558979807099510010550, 14.55847462133591598399845749321, 15.44544303217164622985200394439, 16.18066007274649173202413446868, 18.56744771117491312836050583330, 19.74589105824024445259505976782

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