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Properties

Label 2-13e2-13.9-c3-0-9
Degree $2$
Conductor $169$
Sign $-0.945 - 0.326i$
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  + (−0.866 + 1.5i)2-s + (−1 + 1.73i)3-s + (2.5 + 4.33i)4-s + 1.73·5-s + (−1.73 − 3i)6-s + (6.92 + 12i)7-s − 22.5·8-s + (11.5 + 19.9i)9-s + (−1.49 + 2.59i)10-s + (6.92 − 12i)11-s − 10·12-s − 24·14-s + (−1.73 + 2.99i)15-s + (−0.500 + 0.866i)16-s + (58.5 + 101. i)17-s − 39.8·18-s + ⋯
L(s)  = 1  + (−0.306 + 0.530i)2-s + (−0.192 + 0.333i)3-s + (0.312 + 0.541i)4-s + 0.154·5-s + (−0.117 − 0.204i)6-s + (0.374 + 0.647i)7-s − 0.995·8-s + (0.425 + 0.737i)9-s + (−0.0474 + 0.0821i)10-s + (0.189 − 0.328i)11-s − 0.240·12-s − 0.458·14-s + (−0.0298 + 0.0516i)15-s + (−0.00781 + 0.0135i)16-s + (0.834 + 1.44i)17-s − 0.521·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.326i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.945 - 0.326i$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ -0.945 - 0.326i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.218066 + 1.29911i\)
\(L(\frac12)\) \(\approx\) \(0.218066 + 1.29911i\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.866 - 1.5i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (1 - 1.73i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 - 1.73T + 125T^{2} \)
7 \( 1 + (-6.92 - 12i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-6.92 + 12i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-58.5 - 101. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (57.1 + 99i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (39 - 67.5i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-70.5 + 122. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 155.T + 2.97e4T^{2} \)
37 \( 1 + (71.8 - 124.5i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (135. - 235.5i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-52 - 90.0i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 301.T + 1.03e5T^{2} \)
53 \( 1 - 93T + 1.48e5T^{2} \)
59 \( 1 + (-142. - 246i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (72.5 + 125. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-393. + 681i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-528. - 915i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 458.T + 3.89e5T^{2} \)
79 \( 1 - 1.27e3T + 4.93e5T^{2} \)
83 \( 1 - 789.T + 5.71e5T^{2} \)
89 \( 1 + (488. - 846i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-100. - 174i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.68628611570787835086177279437, −11.71691659146177561836050823119, −10.82377134028855640759825736450, −9.619788816433149196878692530011, −8.452135781620371977430298683931, −7.76964545768307212282114741115, −6.42747578087283493493011287783, −5.40235960257539728770678959734, −3.84851002555649470570630737289, −2.15043839107452258692759141424, 0.68208662283862669500559305946, 1.94098282975302134432456121396, 3.77035506015757454564639316973, 5.44271749200565553006714038086, 6.60603704777648369113821010306, 7.56405279364463047215064457009, 9.132368863614644583100601364512, 10.01288386596610680471340591176, 10.76850912642394784185359377233, 11.96651688181597783949353201126

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