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Properties

Label 2-13e2-13.10-c3-0-33
Degree $2$
Conductor $169$
Sign $-0.967 - 0.252i$
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  + (3.95 − 2.28i)2-s + (−4.34 − 7.52i)3-s + (6.40 − 11.0i)4-s − 2.80i·5-s + (−34.3 − 19.8i)6-s + (−8.28 − 4.78i)7-s − 21.9i·8-s + (−24.2 + 41.9i)9-s + (−6.40 − 11.0i)10-s + (34.1 − 19.7i)11-s − 111.·12-s − 43.6·14-s + (−21.1 + 12.1i)15-s + (1.21 + 2.09i)16-s + (1.00 − 1.74i)17-s + 220. i·18-s + ⋯
L(s)  = 1  + (1.39 − 0.806i)2-s + (−0.835 − 1.44i)3-s + (0.800 − 1.38i)4-s − 0.251i·5-s + (−2.33 − 1.34i)6-s + (−0.447 − 0.258i)7-s − 0.969i·8-s + (−0.896 + 1.55i)9-s + (−0.202 − 0.350i)10-s + (0.935 − 0.540i)11-s − 2.67·12-s − 0.832·14-s + (−0.363 + 0.209i)15-s + (0.0189 + 0.0327i)16-s + (0.0143 − 0.0248i)17-s + 2.89i·18-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.967 - 0.252i$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ -0.967 - 0.252i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.305948 + 2.38291i\)
\(L(\frac12)\) \(\approx\) \(0.305948 + 2.38291i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.95 + 2.28i)T + (4 - 6.92i)T^{2} \)
3 \( 1 + (4.34 + 7.52i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + 2.80iT - 125T^{2} \)
7 \( 1 + (8.28 + 4.78i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-34.1 + 19.7i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-1.00 + 1.74i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (52.1 + 30.0i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-2.23 - 3.87i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (70.3 + 121. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 136. iT - 2.97e4T^{2} \)
37 \( 1 + (160. - 92.8i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (268. - 155. i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-213. + 370. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 258. iT - 1.03e5T^{2} \)
53 \( 1 - 612.T + 1.48e5T^{2} \)
59 \( 1 + (-448. - 258. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-80.6 + 139. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-43.2 + 24.9i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-242. - 139. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 467. iT - 3.89e5T^{2} \)
79 \( 1 - 37.5T + 4.93e5T^{2} \)
83 \( 1 - 76.1iT - 5.71e5T^{2} \)
89 \( 1 + (-175. + 101. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (1.01e3 + 587. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.86438515580381123161869468495, −11.48385880321310453876347423112, −10.36137758610133815812223598287, −8.584696062784225089655533642317, −6.99047648742532076853562136053, −6.23142705559816637303256529916, −5.25317799722920081604531344028, −3.80378183807799651395671400142, −2.18233842210118724626617078097, −0.802982030403776512660976083554, 3.34851378763466217589935829882, 4.27232388456026673565472674511, 5.18739097949554728277277213083, 6.16902062077879287252009606688, 6.99672988952932664938675051264, 8.926933239712475024042906584756, 9.999396599088769190814589731289, 11.01077646583602894659737342849, 12.08806445405784196008583837227, 12.75802411454478172954139830789

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