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Properties

Label 2-13e2-13.4-c3-0-30
Degree $2$
Conductor $169$
Sign $0.252 + 0.967i$
Analytic cond. $9.97132$
Root an. cond. $3.15774$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3 + 1.73i)2-s + (3.5 − 6.06i)3-s + (2 + 3.46i)4-s − 13.8i·5-s + (21 − 12.1i)6-s + (−19.5 + 11.2i)7-s − 13.8i·8-s + (−11 − 19.0i)9-s + (23.9 − 41.5i)10-s + (19.5 + 11.2i)11-s + 28.0·12-s − 78·14-s + (−84 − 48.4i)15-s + (39.9 − 69.2i)16-s + (−13.5 − 23.3i)17-s − 76.2i·18-s + ⋯
L(s)  = 1  + (1.06 + 0.612i)2-s + (0.673 − 1.16i)3-s + (0.250 + 0.433i)4-s − 1.23i·5-s + (1.42 − 0.824i)6-s + (−1.05 + 0.607i)7-s − 0.612i·8-s + (−0.407 − 0.705i)9-s + (0.758 − 1.31i)10-s + (0.534 + 0.308i)11-s + 0.673·12-s − 1.48·14-s + (−1.44 − 0.834i)15-s + (0.624 − 1.08i)16-s + (−0.192 − 0.333i)17-s − 0.997i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.252 + 0.967i$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ 0.252 + 0.967i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.51327 - 1.94133i\)
\(L(\frac12)\) \(\approx\) \(2.51327 - 1.94133i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 + (-3 - 1.73i)T + (4 + 6.92i)T^{2} \)
3 \( 1 + (-3.5 + 6.06i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + 13.8iT - 125T^{2} \)
7 \( 1 + (19.5 - 11.2i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-19.5 - 11.2i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (13.5 + 23.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-76.5 + 44.1i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (28.5 - 49.3i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-34.5 + 59.7i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 72.7iT - 2.97e4T^{2} \)
37 \( 1 + (-34.5 - 19.9i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-340.5 - 196. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-42.5 - 73.6i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 342. iT - 1.03e5T^{2} \)
53 \( 1 - 426T + 1.48e5T^{2} \)
59 \( 1 + (-16.5 + 9.52i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-8.5 - 14.7i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (142.5 + 82.2i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (505.5 - 291. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 1.00e3iT - 3.89e5T^{2} \)
79 \( 1 + 1.24e3T + 4.93e5T^{2} \)
83 \( 1 - 426. iT - 5.71e5T^{2} \)
89 \( 1 + (265.5 + 153. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (1.06e3 - 617. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.56617127475512429808625773718, −11.91807065619107230259362308391, −9.617868604237781245222547209375, −8.977488157071095687247853211859, −7.66488656998015400915562074234, −6.69206949207697721905703745041, −5.71306079556883233103438032691, −4.48477955399350986060343173307, −2.92282485316815195108902246488, −1.05950205786210842519547244835, 2.74425359952995133703088591666, 3.53907287824370339188702844887, 4.16441134363017082980911462640, 5.85313700552246529537196033999, 7.11092459271560226174036489885, 8.693994110293113578780864913178, 9.922501484049223605501861829901, 10.50677118569794995851774029982, 11.45743790207122192673866937264, 12.63721061618998537335721450678

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