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Properties

Label 2-13e2-13.6-c2-0-0
Degree $2$
Conductor $169$
Sign $0.750 - 0.661i$
Analytic cond. $4.60491$
Root an. cond. $2.14590$
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  + (−0.834 − 3.11i)2-s + (1.02 − 1.77i)3-s + (−5.54 + 3.19i)4-s + (−5.50 + 5.50i)5-s + (−6.37 − 1.70i)6-s + (−0.846 + 3.16i)7-s + (5.47 + 5.47i)8-s + (2.40 + 4.16i)9-s + (21.7 + 12.5i)10-s + (−4.90 + 1.31i)11-s + 13.0i·12-s + 10.5·14-s + (4.12 + 15.3i)15-s + (−0.322 + 0.558i)16-s + (−8.72 + 5.03i)17-s + (10.9 − 10.9i)18-s + ⋯
L(s)  = 1  + (−0.417 − 1.55i)2-s + (0.341 − 0.590i)3-s + (−1.38 + 0.799i)4-s + (−1.10 + 1.10i)5-s + (−1.06 − 0.284i)6-s + (−0.120 + 0.451i)7-s + (0.683 + 0.683i)8-s + (0.267 + 0.463i)9-s + (2.17 + 1.25i)10-s + (−0.445 + 0.119i)11-s + 1.09i·12-s + 0.753·14-s + (0.274 + 1.02i)15-s + (−0.0201 + 0.0349i)16-s + (−0.513 + 0.296i)17-s + (0.609 − 0.609i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 - 0.661i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.750 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.750 - 0.661i$
Analytic conductor: \(4.60491\)
Root analytic conductor: \(2.14590\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1),\ 0.750 - 0.661i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.346603 + 0.130927i\)
\(L(\frac12)\) \(\approx\) \(0.346603 + 0.130927i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 + (0.834 + 3.11i)T + (-3.46 + 2i)T^{2} \)
3 \( 1 + (-1.02 + 1.77i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (5.50 - 5.50i)T - 25iT^{2} \)
7 \( 1 + (0.846 - 3.16i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (4.90 - 1.31i)T + (104. - 60.5i)T^{2} \)
17 \( 1 + (8.72 - 5.03i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (11.4 + 3.07i)T + (312. + 180.5i)T^{2} \)
23 \( 1 + (27.5 + 15.9i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (16.3 - 28.2i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (8.05 - 8.05i)T - 961iT^{2} \)
37 \( 1 + (-42.7 + 11.4i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (-12.2 - 45.6i)T + (-1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (20.1 - 11.6i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (64.6 + 64.6i)T + 2.20e3iT^{2} \)
53 \( 1 + 48.7T + 2.80e3T^{2} \)
59 \( 1 + (6.31 - 23.5i)T + (-3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (6.30 + 10.9i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-4.48 - 16.7i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (38.8 + 10.4i)T + (4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (-76.5 - 76.5i)T + 5.32e3iT^{2} \)
79 \( 1 + 94.0T + 6.24e3T^{2} \)
83 \( 1 + (-33.6 + 33.6i)T - 6.88e3iT^{2} \)
89 \( 1 + (35.0 - 9.39i)T + (6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (-40.1 - 10.7i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−12.52723449886998485052934338405, −11.49616873307482845059035463016, −10.83731430973075783277902563943, −10.02810016541315238105503428933, −8.601609711895835011785117062058, −7.79862436917114440173097732412, −6.60506012672403692983224930280, −4.31901653940667426150837757282, −3.05425464923891770394525947691, −2.07533911828733906491818040184, 0.24261133922988061036500242751, 3.92641203491853923619597350671, 4.75801586329778902402611648251, 6.15310656577756503885213410726, 7.50460972570149879862929131429, 8.160358600818152860563275007953, 9.068326832257484902710796694367, 9.879309404989894431069582263308, 11.44461345564866401151999956673, 12.62714290854710913652195072672

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