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Properties

Label 2-13e2-13.8-c4-0-15
Degree $2$
Conductor $169$
Sign $-0.543 + 0.839i$
Analytic cond. $17.4695$
Root an. cond. $4.17965$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.36 − 4.36i)2-s − 5.52·3-s + 22.1i·4-s + (5.37 + 5.37i)5-s + (24.0 + 24.0i)6-s + (−63.9 + 63.9i)7-s + (26.6 − 26.6i)8-s − 50.5·9-s − 46.9i·10-s + (−96.0 + 96.0i)11-s − 122. i·12-s + 558.·14-s + (−29.6 − 29.6i)15-s + 120.·16-s − 112. i·17-s + (220. + 220. i)18-s + ⋯
L(s)  = 1  + (−1.09 − 1.09i)2-s − 0.613·3-s + 1.38i·4-s + (0.215 + 0.215i)5-s + (0.669 + 0.669i)6-s + (−1.30 + 1.30i)7-s + (0.416 − 0.416i)8-s − 0.623·9-s − 0.469i·10-s + (−0.793 + 0.793i)11-s − 0.847i·12-s + 2.84·14-s + (−0.131 − 0.131i)15-s + 0.472·16-s − 0.389i·17-s + (0.680 + 0.680i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.543 + 0.839i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.543 + 0.839i$
Analytic conductor: \(17.4695\)
Root analytic conductor: \(4.17965\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :2),\ -0.543 + 0.839i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.1579981550\)
\(L(\frac12)\) \(\approx\) \(0.1579981550\)
\(L(3)\) 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 \( 1 + (4.36 + 4.36i)T + 16iT^{2} \)
3 \( 1 + 5.52T + 81T^{2} \)
5 \( 1 + (-5.37 - 5.37i)T + 625iT^{2} \)
7 \( 1 + (63.9 - 63.9i)T - 2.40e3iT^{2} \)
11 \( 1 + (96.0 - 96.0i)T - 1.46e4iT^{2} \)
17 \( 1 + 112. iT - 8.35e4T^{2} \)
19 \( 1 + (20.2 + 20.2i)T + 1.30e5iT^{2} \)
23 \( 1 - 441. iT - 2.79e5T^{2} \)
29 \( 1 + 282.T + 7.07e5T^{2} \)
31 \( 1 + (596. + 596. i)T + 9.23e5iT^{2} \)
37 \( 1 + (262. - 262. i)T - 1.87e6iT^{2} \)
41 \( 1 + (-926. - 926. i)T + 2.82e6iT^{2} \)
43 \( 1 - 43.3iT - 3.41e6T^{2} \)
47 \( 1 + (2.36e3 - 2.36e3i)T - 4.87e6iT^{2} \)
53 \( 1 + 150.T + 7.89e6T^{2} \)
59 \( 1 + (-1.22e3 + 1.22e3i)T - 1.21e7iT^{2} \)
61 \( 1 + 2.13e3T + 1.38e7T^{2} \)
67 \( 1 + (-3.94e3 - 3.94e3i)T + 2.01e7iT^{2} \)
71 \( 1 + (5.09e3 + 5.09e3i)T + 2.54e7iT^{2} \)
73 \( 1 + (-3.85e3 + 3.85e3i)T - 2.83e7iT^{2} \)
79 \( 1 - 9.01e3T + 3.89e7T^{2} \)
83 \( 1 + (-2.18e3 - 2.18e3i)T + 4.74e7iT^{2} \)
89 \( 1 + (-987. + 987. i)T - 6.27e7iT^{2} \)
97 \( 1 + (-5.00e3 - 5.00e3i)T + 8.85e7iT^{2} \)
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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

−11.67038703788917120969244525405, −10.69374606608972391369088352017, −9.707191322932067180876235170419, −9.188619214856716715226878786736, −7.953047527137987165277691097604, −6.36318611298417149293967956850, −5.37640240886438340840778831717, −3.08292009454071813188451458946, −2.24472420633326292840605989939, −0.15506775434369160441470462788, 0.66536196477165957923388554697, 3.42415503940599995760896788915, 5.42409263877534343640969413580, 6.32408575913150351480539102119, 7.14811050585822988323247687174, 8.278334029831056103072861706176, 9.278356648618855144363186289947, 10.32070359885443436760563636952, 10.91177586946990780905727307751, 12.58245238774226819516722232526

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