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Properties

Label 2-13e2-13.4-c3-0-16
Degree $2$
Conductor $169$
Sign $-0.542 - 0.839i$
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  + (4.70 + 2.71i)2-s + (−0.837 + 1.45i)3-s + (10.7 + 18.5i)4-s − 7.70i·5-s + (−7.87 + 4.54i)6-s + (−13.0 + 7.51i)7-s + 73.1i·8-s + (12.0 + 20.9i)9-s + (20.9 − 36.2i)10-s + (2.17 + 1.25i)11-s − 35.9·12-s − 81.5·14-s + (11.1 + 6.45i)15-s + (−112. + 194. i)16-s + (−1.03 − 1.78i)17-s + 131. i·18-s + ⋯
L(s)  = 1  + (1.66 + 0.959i)2-s + (−0.161 + 0.279i)3-s + (1.34 + 2.32i)4-s − 0.689i·5-s + (−0.535 + 0.309i)6-s + (−0.702 + 0.405i)7-s + 3.23i·8-s + (0.448 + 0.776i)9-s + (0.661 − 1.14i)10-s + (0.0595 + 0.0344i)11-s − 0.865·12-s − 1.55·14-s + (0.192 + 0.111i)15-s + (−1.75 + 3.04i)16-s + (−0.0147 − 0.0255i)17-s + 1.71i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.542 - 0.839i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.542 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.542 - 0.839i$
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: \((2,\ 169,\ (\ :3/2),\ -0.542 - 0.839i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.87389 + 3.44282i\)
\(L(\frac12)\) \(\approx\) \(1.87389 + 3.44282i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.70 - 2.71i)T + (4 + 6.92i)T^{2} \)
3 \( 1 + (0.837 - 1.45i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + 7.70iT - 125T^{2} \)
7 \( 1 + (13.0 - 7.51i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-2.17 - 1.25i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (1.03 + 1.78i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-81.7 + 47.2i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-17.9 + 31.0i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-70.0 + 121. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 264. iT - 2.97e4T^{2} \)
37 \( 1 + (222. + 128. i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-341. - 197. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-128. - 222. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 415. iT - 1.03e5T^{2} \)
53 \( 1 + 504.T + 1.48e5T^{2} \)
59 \( 1 + (-104. + 60.1i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-376. - 651. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-183. - 105. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-355. + 205. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 17.4iT - 3.89e5T^{2} \)
79 \( 1 + 174.T + 4.93e5T^{2} \)
83 \( 1 + 963. iT - 5.71e5T^{2} \)
89 \( 1 + (413. + 238. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (906. - 523. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.98156548814193165864364654004, −12.11816927241830408943813986579, −11.12894575132034979887503521685, −9.492235534789599649993515216632, −8.141903353631335339781854415797, −7.12775832859084613411419522576, −5.95825632281372819863485309209, −5.05197369986610444378690104014, −4.16295105478366612052078413812, −2.68705197793387958838590065206, 1.23700543535223819638544282773, 3.03211721797354545617870189231, 3.76750089951273494318081626355, 5.24878199553386690227466276085, 6.48576367953201415051373948764, 7.04767560277173963115490359313, 9.521028954342913816032151703392, 10.40035146141628401697988600135, 11.20025479321438675484723323065, 12.37422263839907104378371274920

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