Properties

Label 2-13e2-13.4-c3-0-32
Degree $2$
Conductor $169$
Sign $-0.802 - 0.596i$
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  + (0.129 + 0.0745i)2-s + (3.24 − 5.61i)3-s + (−3.98 − 6.90i)4-s + 10.2i·5-s + (0.837 − 0.483i)6-s + (−25.6 + 14.8i)7-s − 2.38i·8-s + (−7.55 − 13.0i)9-s + (−0.764 + 1.32i)10-s + (−32.9 − 19.0i)11-s − 51.7·12-s − 4.42·14-s + (57.6 + 33.2i)15-s + (−31.7 + 54.9i)16-s + (−35.6 − 61.7i)17-s − 2.25i·18-s + ⋯
L(s)  = 1  + (0.0456 + 0.0263i)2-s + (0.624 − 1.08i)3-s + (−0.498 − 0.863i)4-s + 0.917i·5-s + (0.0569 − 0.0329i)6-s + (−1.38 + 0.801i)7-s − 0.105i·8-s + (−0.279 − 0.484i)9-s + (−0.0241 + 0.0418i)10-s + (−0.904 − 0.522i)11-s − 1.24·12-s − 0.0844·14-s + (0.991 + 0.572i)15-s + (−0.495 + 0.858i)16-s + (−0.508 − 0.880i)17-s − 0.0294i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.802 - 0.596i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.802 - 0.596i$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ -0.802 - 0.596i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0394996 + 0.119447i\)
\(L(\frac12)\) \(\approx\) \(0.0394996 + 0.119447i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.129 - 0.0745i)T + (4 + 6.92i)T^{2} \)
3 \( 1 + (-3.24 + 5.61i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 - 10.2iT - 125T^{2} \)
7 \( 1 + (25.6 - 14.8i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (32.9 + 19.0i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (35.6 + 61.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (8.74 - 5.04i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (99.3 - 172. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (15.4 - 26.6i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 151. iT - 2.97e4T^{2} \)
37 \( 1 + (131. + 75.6i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (179. + 103. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (151. + 262. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 12.2iT - 1.03e5T^{2} \)
53 \( 1 + 250.T + 1.48e5T^{2} \)
59 \( 1 + (-338. + 195. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-78.2 - 135. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (263. + 151. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (791. - 456. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 249. iT - 3.89e5T^{2} \)
79 \( 1 + 147.T + 4.93e5T^{2} \)
83 \( 1 + 1.02e3iT - 5.71e5T^{2} \)
89 \( 1 + (-819. - 473. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (361. - 208. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.87498869572690177627234260347, −10.56612986706277597226919040094, −9.642332852999105282443149186475, −8.684807011431600042428256702247, −7.35896304336122111223001609233, −6.43655463550733485133824773056, −5.49195926095208368339458665862, −3.27129695174990985054150670272, −2.18471356752489482811119463321, −0.04980980498840842621380763588, 2.98180667248941052317198166115, 4.05883700330168856838472399786, 4.77047707732924287736455333331, 6.70202012119615212957538999719, 8.173793543970602710288374316630, 8.869546657155417516321815756509, 9.867539948207911932418137730909, 10.47029862344170271398827827243, 12.37973685421506420848011142604, 12.88235232443863894527394030312

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