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Properties

Label 2-13e2-13.5-c4-0-40
Degree $2$
Conductor $169$
Sign $0.0129 + 0.999i$
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  + (4.03 − 4.03i)2-s + 8.56·3-s − 16.5i·4-s + (29.3 − 29.3i)5-s + (34.5 − 34.5i)6-s + (33.5 + 33.5i)7-s + (−2.12 − 2.12i)8-s − 7.69·9-s − 236. i·10-s + (−31.3 − 31.3i)11-s − 141. i·12-s + 270.·14-s + (251. − 251. i)15-s + 247.·16-s + 237. i·17-s + (−31.0 + 31.0i)18-s + ⋯
L(s)  = 1  + (1.00 − 1.00i)2-s + 0.951·3-s − 1.03i·4-s + (1.17 − 1.17i)5-s + (0.959 − 0.959i)6-s + (0.684 + 0.684i)7-s + (−0.0332 − 0.0332i)8-s − 0.0950·9-s − 2.36i·10-s + (−0.258 − 0.258i)11-s − 0.982i·12-s + 1.38·14-s + (1.11 − 1.11i)15-s + 0.965·16-s + 0.821i·17-s + (−0.0958 + 0.0958i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0129 + 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0129 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.0129 + 0.999i$
Analytic conductor: \(17.4695\)
Root analytic conductor: \(4.17965\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (70, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :2),\ 0.0129 + 0.999i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(5.206772519\)
\(L(\frac12)\) \(\approx\) \(5.206772519\)
\(L(3)\) 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 \( 1 + (-4.03 + 4.03i)T - 16iT^{2} \)
3 \( 1 - 8.56T + 81T^{2} \)
5 \( 1 + (-29.3 + 29.3i)T - 625iT^{2} \)
7 \( 1 + (-33.5 - 33.5i)T + 2.40e3iT^{2} \)
11 \( 1 + (31.3 + 31.3i)T + 1.46e4iT^{2} \)
17 \( 1 - 237. iT - 8.35e4T^{2} \)
19 \( 1 + (348. - 348. i)T - 1.30e5iT^{2} \)
23 \( 1 + 298. iT - 2.79e5T^{2} \)
29 \( 1 + 371.T + 7.07e5T^{2} \)
31 \( 1 + (625. - 625. i)T - 9.23e5iT^{2} \)
37 \( 1 + (540. + 540. i)T + 1.87e6iT^{2} \)
41 \( 1 + (491. - 491. i)T - 2.82e6iT^{2} \)
43 \( 1 + 1.12e3iT - 3.41e6T^{2} \)
47 \( 1 + (609. + 609. i)T + 4.87e6iT^{2} \)
53 \( 1 - 897.T + 7.89e6T^{2} \)
59 \( 1 + (-351. - 351. i)T + 1.21e7iT^{2} \)
61 \( 1 - 248.T + 1.38e7T^{2} \)
67 \( 1 + (-1.31e3 + 1.31e3i)T - 2.01e7iT^{2} \)
71 \( 1 + (2.12e3 - 2.12e3i)T - 2.54e7iT^{2} \)
73 \( 1 + (-6.81e3 - 6.81e3i)T + 2.83e7iT^{2} \)
79 \( 1 - 2.50e3T + 3.89e7T^{2} \)
83 \( 1 + (7.97e3 - 7.97e3i)T - 4.74e7iT^{2} \)
89 \( 1 + (-1.02e4 - 1.02e4i)T + 6.27e7iT^{2} \)
97 \( 1 + (-5.23e3 + 5.23e3i)T - 8.85e7iT^{2} \)
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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

−12.27978675463812948810908083425, −10.99639725993615590245830872145, −9.938475814975867587118227077293, −8.688522665555702625043784630735, −8.287092048967366251736938011315, −5.84234706122467137822923850013, −5.13424709217623389252515411675, −3.82395741604106196014581282472, −2.30871076451037160344856366767, −1.68744934118681677849763308474, 2.13694744208956588209170342767, 3.37178788237383706114548595985, 4.81382345788696683328901273068, 5.99322411307710491199144355966, 7.03560396663411229575655385596, 7.74777814462278104054786355258, 9.210848828884918042717118250972, 10.30402739173812940256773354111, 11.28618123786080989500425185198, 13.16982266558798196404240479064

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