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Properties

Label 2-13e2-13.3-c3-0-7
Degree $2$
Conductor $169$
Sign $-0.309 - 0.950i$
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  + (1.58 + 2.73i)2-s + (−3.54 − 6.13i)3-s + (−0.997 + 1.72i)4-s − 13.6·5-s + (11.2 − 19.4i)6-s + (−7.16 + 12.4i)7-s + 18.9·8-s + (−11.6 + 20.1i)9-s + (−21.5 − 37.2i)10-s + (33.8 + 58.6i)11-s + 14.1·12-s − 45.3·14-s + (48.2 + 83.5i)15-s + (37.9 + 65.7i)16-s + (−0.168 + 0.292i)17-s − 73.5·18-s + ⋯
L(s)  = 1  + (0.558 + 0.967i)2-s + (−0.682 − 1.18i)3-s + (−0.124 + 0.215i)4-s − 1.21·5-s + (0.762 − 1.32i)6-s + (−0.386 + 0.670i)7-s + 0.839·8-s + (−0.430 + 0.745i)9-s + (−0.679 − 1.17i)10-s + (0.928 + 1.60i)11-s + 0.340·12-s − 0.864·14-s + (0.829 + 1.43i)15-s + (0.593 + 1.02i)16-s + (−0.00240 + 0.00417i)17-s − 0.962·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.309 - 0.950i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.309 - 0.950i$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ -0.309 - 0.950i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.692687 + 0.953946i\)
\(L(\frac12)\) \(\approx\) \(0.692687 + 0.953946i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.58 - 2.73i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (3.54 + 6.13i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + 13.6T + 125T^{2} \)
7 \( 1 + (7.16 - 12.4i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-33.8 - 58.6i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (0.168 - 0.292i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (20.2 - 35.0i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-77.8 - 134. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-16.8 - 29.2i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 157.T + 2.97e4T^{2} \)
37 \( 1 + (-29.3 - 50.7i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (29.6 + 51.3i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-104. + 180. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 221.T + 1.03e5T^{2} \)
53 \( 1 + 409.T + 1.48e5T^{2} \)
59 \( 1 + (86.7 - 150. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (280. - 485. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-134. - 233. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (30.4 - 52.8i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 282.T + 3.89e5T^{2} \)
79 \( 1 - 984.T + 4.93e5T^{2} \)
83 \( 1 - 1.20e3T + 5.71e5T^{2} \)
89 \( 1 + (-269. - 467. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (793. - 1.37e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.44423933890769324772949803769, −12.09183278439736254376896418203, −11.03710570690334846714235912080, −9.407776691346512177441807137943, −7.81964894010515806698840080902, −7.16974998828886022943151388180, −6.46755597882488392520340642372, −5.31959947716326335784430926109, −4.00970907460292610447460464367, −1.59203785230204965261393429457, 0.52243126721513021669868818410, 3.33709390935274612056950889583, 3.93180304815881742172388046337, 4.85109010167840236352875507493, 6.50888087201839680824291913402, 7.961237326732023995715243045234, 9.292290548761044842283449637189, 10.66389851580807739704785657952, 11.05113255367863619053347690216, 11.68165925969317224815209892307

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