Properties

Label 2-13e2-169.17-c1-0-5
Degree $2$
Conductor $169$
Sign $0.369 - 0.929i$
Analytic cond. $1.34947$
Root an. cond. $1.16166$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.52 + 0.514i)2-s + (2.70 + 0.439i)3-s + (4.24 − 1.81i)4-s + (0.847 + 3.43i)5-s + (−7.05 + 0.284i)6-s + (0.693 + 0.328i)7-s + (−5.54 + 3.82i)8-s + (4.28 + 1.43i)9-s + (−3.90 − 8.22i)10-s + (−1.32 − 3.95i)11-s + (12.2 − 3.03i)12-s + (−2.08 − 2.94i)13-s + (−1.91 − 0.472i)14-s + (0.781 + 9.67i)15-s + (5.60 − 5.83i)16-s + (−2.29 + 4.83i)17-s + ⋯
L(s)  = 1  + (−1.78 + 0.363i)2-s + (1.56 + 0.254i)3-s + (2.12 − 0.905i)4-s + (0.378 + 1.53i)5-s + (−2.87 + 0.115i)6-s + (0.262 + 0.124i)7-s + (−1.95 + 1.35i)8-s + (1.42 + 0.477i)9-s + (−1.23 − 2.60i)10-s + (−0.398 − 1.19i)11-s + (3.55 − 0.875i)12-s + (−0.576 − 0.816i)13-s + (−0.512 − 0.126i)14-s + (0.201 + 2.49i)15-s + (1.40 − 1.45i)16-s + (−0.556 + 1.17i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.369 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.369 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.369 - 0.929i$
Analytic conductor: \(1.34947\)
Root analytic conductor: \(1.16166\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1/2),\ 0.369 - 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.738024 + 0.500961i\)
\(L(\frac12)\) \(\approx\) \(0.738024 + 0.500961i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (2.08 + 2.94i)T \)
good2 \( 1 + (2.52 - 0.514i)T + (1.83 - 0.783i)T^{2} \)
3 \( 1 + (-2.70 - 0.439i)T + (2.84 + 0.950i)T^{2} \)
5 \( 1 + (-0.847 - 3.43i)T + (-4.42 + 2.32i)T^{2} \)
7 \( 1 + (-0.693 - 0.328i)T + (4.42 + 5.42i)T^{2} \)
11 \( 1 + (1.32 + 3.95i)T + (-8.79 + 6.60i)T^{2} \)
17 \( 1 + (2.29 - 4.83i)T + (-10.7 - 13.1i)T^{2} \)
19 \( 1 + (-1.08 + 0.625i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.81 + 3.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.242 - 1.18i)T + (-26.6 + 11.3i)T^{2} \)
31 \( 1 + (-1.61 + 3.08i)T + (-17.6 - 25.5i)T^{2} \)
37 \( 1 + (-5.46 - 8.63i)T + (-15.8 + 33.4i)T^{2} \)
41 \( 1 + (-1.11 + 6.83i)T + (-38.8 - 12.9i)T^{2} \)
43 \( 1 + (4.94 + 3.12i)T + (18.4 + 38.8i)T^{2} \)
47 \( 1 + (-1.77 + 0.215i)T + (45.6 - 11.2i)T^{2} \)
53 \( 1 + (-0.439 - 0.636i)T + (-18.7 + 49.5i)T^{2} \)
59 \( 1 + (3.29 - 3.16i)T + (2.37 - 58.9i)T^{2} \)
61 \( 1 + (-7.51 - 0.606i)T + (60.2 + 9.78i)T^{2} \)
67 \( 1 + (-3.22 + 7.57i)T + (-46.4 - 48.3i)T^{2} \)
71 \( 1 + (7.05 + 5.75i)T + (14.2 + 69.5i)T^{2} \)
73 \( 1 + (7.64 - 8.62i)T + (-8.79 - 72.4i)T^{2} \)
79 \( 1 + (-0.238 - 1.96i)T + (-76.7 + 18.9i)T^{2} \)
83 \( 1 + (-1.92 - 0.731i)T + (62.1 + 55.0i)T^{2} \)
89 \( 1 + (7.96 + 4.59i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.85 + 0.827i)T + (81.9 - 51.8i)T^{2} \)
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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

−13.31974093151996889973393178938, −11.26582800930131086140518377187, −10.44289220405895075885572777661, −9.915687513169550550466849769448, −8.694583103312550327494451012327, −8.143981709367129058840303024707, −7.20916929314645384994084711491, −6.10721768411212323166919766461, −3.14935214784112573783235061802, −2.28805317398735887105919647157, 1.51413618322468157767354325780, 2.51272053170080896808376979212, 4.65189185393078644688745264470, 7.13977966595557185307406924661, 7.81927221090528229570266208284, 8.753941108776478581823528495350, 9.452337240846122867015363356073, 9.776550951459245628348494462943, 11.50948369861076278476728389264, 12.52939881061302229783767246956

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